This is a declarative existential clause with the dummy subject there and the copular auxiliary be taking a noun phrase (NP) complement:
There is [ nothing a guy can do that even comes close ]
We can, of course, divide the NP up further. Its head is nothing, which is modified by two relative clauses stacked together:
nothing [ Ø a guy can do ____ ] [ that ____ even comes close ]
Each relative clause contains a gap, indicated above with underscore notation
____. In the first clause the gap is in object position, and in the second clause the gap is in subject position. In both cases, the gap is coreferential with the head nothing.
In neither case is there an overt anaphoric link between the head and the gap in the form of a relative word. (That in this sentence would traditionally be a considered relative pronoun rather than a complementizer, but the distinction is unimportant here so I won't address it).
Let's look at the individual relative clauses:
[ Ø a guy can do ____ ]
Here we have a bare relative with neither the complementizer that nor the relative pronoun which. This is possible for two reasons:
- Bare relatives are permitted only when the gap is in non-subject position.
- When relative clauses are stacked, only the first can be a bare relative.
Both conditions here are met, so a bare relative is allowed.
[ that ____ even comes close ]
The second relative is a that-relative, introduced by the complementizer that. The same two reasons prevent it from appearing as a bare relative:
- The gap is in subject position.
- This is a non-initial stacked relative.
As a result, either the complementizer that or the relative pronoun which must be used.
Since neither relative contains an overt wh-word and neither relative is set off by commas, we can see that both are integrated relatives and should be pronounced as part of a larger intonational unit. (You may also call these by the traditional term restrictive relative clause, if you prefer—it makes no difference in this particular case.)