Learn French: Vous pouvez faire toute la réparation vous-même.

It acts as an emphatic pronoun. While vous just means you, vous-même emphasizes that you can do it yourself, meaning in person or without anyone else's help.

Does vous-même tell us if the speaker is talking to one person or multiple people?

Yes, it tells us they are talking to exactly one person using the formal vous. If they were talking to a group of people, it would be spelled vous-mêmes with an s at the end.

Why is it toute la réparation and not just toute réparation?

When you use tout or its feminine form toute followed immediately by a definite article like la, it means the whole or all of the. So toute la réparation means the whole repair. If you dropped the la and said toute réparation, it would change the meaning to every repair or any repair.

Could I put vous-même earlier in the sentence, like right after pouvez?

No, emphatic pronouns like vous-même usually go at the very end of the clause or sentence to reinforce the subject. Placing it right after pouvez or faire would sound unnatural here.