I was thinking of making a character that has no hands, and was wondering if there's any way to get around somatic components of spells. I made a list of all the spells that don't have somatic components, and they're not that great, mostly.
Ask your DM
I know it looks like a catch all answer, but this time there are very important reasons to do this.
There are hardly any rules for characters without hands
Your DM may not feel OK with additional workload caused by a character rules don't expect to appear at the game table. What about checks like climb? Swim? Using wand or manipulating material components? Thieves cant with its gestures? He would constantly need to make rulings just because of you. This may slow down the game and steal his enjoyment. It can move his attention away from other players.
You shouldn't invest too much time in developing that concept unless you can reasonably expect it might be allowed at your table.
You will be playing on rulings anyway
Thus, if your DM is willing to accommodate your character, he may well decide to make a ruling that he simply can ignore somatic components and does not need a hand to manipulate material components. Sounds like a fair trade for all the things your character will never be able to do.
A third-level sorceror can use the metamagic ability "subtle spell" to cast a spell without any verbal or somatic components. This consumes sorcery points which could have been used for other things instead, so it's not without cost.
I'm not aware of any other ways to remove somatic components from spells.
Incidentally, the best class for a handless character is probably monk, since monks fight with unarmed strikes and unarmed strikes can be kicks.
How about the creative use of a Prosthetic Limb or natural appendages?
MoonMoon, my poor druid who got his hand chopped off in Tomb of Horrors, has recently obtained a prosthetic hand (Wayfinder's Guide to Eberron) to be able to wield a shield and his staff simultaneously. If you really want your character to remain handless, you could work with your DM to obtain a limb that is considered valid for somatic components, but doesn't work as a proper hand. It is a magical item requiring attunement, so not an easy price to pay.
You can also consider using the natural appendages of some races. The Loxodon has a trunk, the Simic has tentacles/claws, and there are probably others I don't recall. You can work with your DM on whether they'd allow this.
Other options include having others casting the spell for you (scrolls, familiars, glyphs, etc), or simply working with your DM to be able to perform somatic components with your arms instead of your hands, as long as they aren't busy with other stuff (stapled to a shield, for example).
I know it's not based on the D&D 5e rules, but in Forgotten Realms lore, there is at least one mage able to cast spells with his feet, when having his hands tied - Pharaun Mizzrym:
Pharaun [...] was skilled in other forms of magic as well; he was even able to cast a spell with his feet.
Emphasis mine. The article cites p. 298 of the Richard Lee Byers novel Dissolution (Book 1 in the War of the Spider Queen series) as the source of this information.
So, if you need to depend on DM fiat/rulings, as the highest-voted answer suggests, that could be a nice way of doing it.