My question is mostly in the title, but here's an example.
Jeff, a STR 19 Bear Totem Barbarian wants cover as he advances towards enemies. To do so, he rips a heavy wooden door off its hinges, and fashions it with straps to be held with two hands as a full-body shield.
Jeff can carry 285lb and push 570lb without encumbrance. Would he be able to benefit from full cover while standing behind his shield?
There are no rules about improvised shields and there are no shields that provide full cover in 5e.
It will be entirely up to the DM as to whether or not they allow the homebrew of a Tower Shield and how such an item would work mechanically.
SRD never rules out the possibility to take your full cover with you.
That would be the end of it, as improvised shields are ultimately up to the DMs discretion, but consider the following:
Doors and other objects that are often larger than a human, thus granting full cover, are unwieldy to carry at best. Jeff (who I'll assume is an average-sized Human) is possibly a few inches shorter than the door itself, thus he'll have to carry it at an angle, lift it off the ground or otherwise really play around with his unwieldy, awkward mobile cover. Depending on the door, it would be better considered 3/4th cover since you have to expose some of your body to not drag it along the floor.
Additionally, a Fragile, Wooden door would have 1d10 HP (while a Resilient door would have 27 HP) which can take just a few attacks before shattering completely if you consider Jeff didn't damage it by pulling it off the structure.
As always, your DM will have the final say in all such matters, and any rules we state are subject to their rulings. That being said, the rules for total cover state:
A target has total cover if it is completely concealed by an obstacle. (PHB, p. 196, bold added)
A definition of "obstacle" is:
a thing that blocks one's way or prevents or hinders progress. (Google Dictionary)
An object you are wielding couldn't possibly block your progress: you're carrying it with you. So such an object is not an obstacle.
You could argue that the "obstacle" is not meant to block progress of a person, but rather of an attack. But by this definition, any person who is wearing armor which completely conceals their skin with anything designed to impede attacks (such as thick quilted padding) would be considered to have total cover and "can’t be targeted directly by an attack or a spell." This would make any fully armored character utterly immune to most spells and attacks.
It is reasonable to demand why a door you are carrying wouldn't give full cover, but a closed door would. Essentially, it's because of the same reason that one is an obstacle and the other is not: because a closed door isn't just a door, it's part of a house.
When a person is trying to break down a door, they are usually not trying to dismantle the door entirely, but rather to dislodge it from its frame. The frame transfers energy of attack delivered to the door into the wall around it, and makes it much harder to move. If a door is unhinged from a wall, though it remains a strong material, it is much easier to move from side to side, and cannot impede attacks as well. At best, it functions as a shield, not as an obstacle.
NOTE: A sufficiently strong character might be able to push or drag an object along that provided them with cover. But this is different than "wielding" it. And most objects that could provide total cover (and are thus heavy enough to serve as obstacles by themselves, unattached to some larger structure, and long enough to completely cover your space) would likely be heavier than your carrying capacity (and reduce your speed to 5 feet via the rules on dragging and carrying in the PHB, p. 176).
To get a comparison of how difficult cover is to obtain while under your control, take a look at Bigby's Hand:
5th-level evocation
Duration: Concentration for up to 1 minute
You create a Large hand of force in an unoccupied space that you can see. It moves at your command, mimicking the movements of your own hand.
The hand is an object that has AC 20 and hit points equal to your hit point maximum. If it drops to 0 hp, the spell ends. It has a Strength of 26 and a Dexterity of 10. The hand doesn't fill its space.
Interposing Hand: The hand places itself between you and a creature of your choice until you give it a different command. It moves to stay between you and the target, providing you with half cover. The target can't move through the hand if its Strength is less than or equal to the hand's. If its score is higher than the hand's, the target can move toward you through the hand, but the space is difficult terrain.
Considering this is a level 5 spell, the caster is likely level 9+, with about 36+ health. The Hand will have the same amount of health. The Hand can only provide Half Cover, at the cost of Concentration, and at the cost of a level 5 spell.
Now, Bigby's Hand does several other things, which definitely contributes to the value of the spell, but few other examples exist that provide any form of cover.
Providing portable full cover with no time duration is likely as powerful as a level 2-3 spell, and making that available as a repeatable improvised action is a bit too strong.
To compensate, I'd likely say that utilizing such a bulky improvised device requires a lot of control, and so he has disadvantage to attack while benefiting from its cover. He could probably slam it into the ground to make it keep its position, but I'd require it to consume his action to do so. Lastly, I'd make it have about 1d12 HP with vulnerability to Fire.
It is a cool idea, and it should be rewarded, but it should always fall short behind professionally made objects or spells that can provide similar options. This is a door that was ripped off a house. For a less hindering and more durable version, he should probably look into a tower shield of some sort.
Like all things, you should wonder if this makes sense in the story you and your players are trying to tell. What kind of person is Jeff? Because if the player, playing with Jeff just wants to abuse the mechanic of full cover, this isn't really telling a compelling story.
If Jeff has a background in war where he saw 80% of his fighting companions slowly being picked of by volleys of arrows and he still has nightmares about it. Now this could make some fun story telling.
Jeff either needs to fix the leather straps to the door too. Achieving is quite difficult, as these straps should hold the weight of the door without hurting your hands/arms like crazy.
So that would require a hard DC leather working INT check to make that happen. You can add more depth to allow any score to be rolled and the difference between DC 20 and the score is the amount of ours the leather straps hold and they simply break, one after the other after such time.
I chose INT ability as you need to understand how the mechanics work and make sure the leather will stay fixed
Yet gaining full cover (and thus immune for targeted attacks) should come at a hefty price.
So for the fun of it I checked on the densities of wood. So presuming the door is solid and roughly has the size of 3ft x 7 ft and 4 inches thick. Depending on the type of wood, it should weigh between 140 lbs and 280 lbs.
An solid Oaken door would weigh 280 lbs and I would reckon about 4d10 HP. A solid Cedar door would weigh 140 lbs and I would give it abot 2d10 HP.
So you can see that carrying some gear together with the door would become cumbersome, like it should be.
Pushing a door around would make it flip over, so not very practical.
This is just spit-balling to get toward something that might work at a price but yet gives significant advantage. I would allow it with the following categories:
Light door: 2d10 (11) HP, AC 14 and you move as if Encumbered, weighs 140 lbs and has DC 15 to attach leather straps to it.
Heavy door: 4d10 (22) HP, AC 16 and you move as if Heavily Encumbered weighs 280 lbs and has DC 20 to attach leather straps to it.
The reasons for encumbrance are the weight and the difficulty to effectively move with the door and staying in it's cover.
Being encumbered by the weight on to of carrying this wooden shield stacks an extra penalty of 10 ft to you speed.
I would give Wooden shield doors:
And have the Two-handed and Heavy properties.
