Does this sentence make sense?

Yes it makes sense as it is.

"There's nothing that you can't do" means that "You can do anything".

"There's something that you can't do" means that "You can't do everything".

The sentence makes perfect sense and is fine.

If you change the word 'nothing' for 'something' it changes the sentence completely.

Your sentence is saying you can do 'anything' whereas the word 'something' say there are some things you can't do.

It makes perfect sense, meaning no one should tell you there are no options available for you.

"There's nothing you can't do" means that you are capable of achieving anything, or whatever you want. Somebody might well say that to someone who is lacking confidence.

Your sentence is saying "Don't let anybody tell you that," which means the opposite - so you can't say it doesn't make sense, but it runs counter to the general feeling of encouragement implied in the first sentence. So your suggestion of "something" would work there, though it would sound a bit of a muddle. "Anything" would be better than "something." But what are you wanting it to mean?

It should be "something", assuming you mean "Never let anyone tell you that you cannot do something."

If, however, you MEAN "Never let anyone tell you that you can do anything you want", then "nothing" is correct.

no

yes, it doesn't technically make sense. that's a double negative. So that sentence is like saying "Never let anyone tell you that there's nothing you CAN do" because saying "there's nothing you can't do" is saying you can do everything