Learn French: Ils portent tout le bois eux-mêmes sans aucune aide. - They carry all the wood themselves without any help.

The suffix -même (meaning "self") must agree in number with the pronoun it attaches to. Since eux (them) is plural, même takes an s. You must also always use a hyphen to connect a stressed pronoun (like moi, toi, lui, nous, vous, eux, elles) to -même.

Could we place eux-mêmes at the very end of the sentence?

You have some flexibility, but putting it at the very end (sans aucune aide eux-mêmes) sounds awkward because it is separated too far from the subject Ils. Placing it right after the object (tout le bois eux-mêmes) is the most natural way to emphasize that they did the carrying themselves. You could also safely say Ils portent eux-mêmes tout le bois.

Why do we use aucune here instead of quelque or n'importe quelle to mean "any"?

The preposition sans (without) carries a negative meaning in French. When you want to say "without any [noun]", you use sans aucun or sans aucune. Because sans already provides the negative context, you drop the ne that normally pairs with aucun. Since aide is feminine, it becomes sans aucune aide.