Your test says it's incorrect, so you'd better say it's incorrect when you're asked.
But I think it's arguably grammatical. We can interpret even though unfit as something called a verbless clause. In this sort of clause, the verb is deleted, and commonly the subject is as well; the verb is generally a form of be, and the subject (when deleted) is recoverable from the main clause.
Here are some examples from Quirk et al's 1985 grammar:
Though already middle-aged, he took a swim every morning.
Though [he was] already middle-aged, he took a swim every morning.
Whether right or wrong, the government always wins the argument.
Whether [it is] right or wrong, the government always wins the argument.
We can do the same thing with your example:
Even though unfit, I still was able to get to the top of the mountain.
Even though [I am] unfit, I still was able to get to the top of the mountain.
Whether the form of be should be past or present probably depends on the context of the sentence. I chose the present tense am here based on the other examples.
When we start talking about missing words, we're talking about theory rather than fact. The missing words aren't actually there, of course! We're just trying to come up with the simplest explanation for why these sentences work when they do.
So there's no requirement that you analyze these as having missing words. But however you choose to analyze them, you'll have to at least understand them when other people say them. And whether or not you decide this particular example is grammatical (I think it is, but another speaker might not), you'll have to admit that other verbless clauses are grammatical.
Here's one last example, this time courtesy of Huddleston & Pullum:
While in Boston I lived with my aunt.
While [I was] in Boston I lived with my aunt.
This sort of sentence is so common that I don't think you could possibly call it ungrammatical.
In any case, best of luck with your test!