An arcane focus can be a very small item. For a War Caster Eldritch Knight, I'd rule that you can wear your arcane focus, that you can hold it in your shield hand on the inside of your shield, that it could be a rune carved in your shield, or even that your weapon or shield could serve as the focus.
For example, in one campaign I'm running, we have a War Priest who is also a War Caster. He literally uses his sword as his holy symbol. I think that is very much in the spirit of RAI for War Caster.
If your DM is particularly picky on RAW, you could see if they allow you to have your arcane focus (e.g. a crystal or other small object) mounted on your shield or weapon. Work with your DM to figure out the cost of doing this. Strictly speaking a War Caster needs to be able to manipulate the arcane focus, but does not otherwise need to use their hands for spellcasting. So that might be a way around.
Do note that any character, War Caster or not, can sheathe their weapon, cast their spell, and then use the next turn's interaction to draw the weapon again. Failing that, a strict reading of RAW says you can drop your weapon for free. Then you can cast your spell as your action and use your "one object interaction" to pick up your weapon again on the same turn. For these reasons, I usually don't bother keeping track of item juggling for spellcasters in my campaigns, as it serves little purpose other than slowing the game down.
Then you ask "what is the point of the War Caster feat, since any spellcaster can manipulate the action economy in this way?" Well, besides the fact that the feat gives you an advantage on saving throws to keep concentrating on spells in the face of damage, and it grants the ability to cast spells as opportunity attacks, it lets you cast a somatic spell with both hands full of weapons/shields in circumstances where sheathing/dropping your weapon is not an option.
For example, War Caster Eldritch Knights can be holding a shield, draw their weapon, attack with that weapon, and then use an action surge to cast a somatic spell, all on the same turn. They can also use a reaction to cast a spell (opportunity attacks or prepared actions), with weapons in both hands. Quite a feat.