Yes, this is a relative construction with a gap directly following the verb in complement position. However, it is somewhat unusual in that its head is a superlative adjective rather than a noun.
Superlative phrases with relative modification do not permit wh-relatives, only bare or that-relatives:
1a. [ the closest that I've come ____ to quitting my job ]
1b. [ the closest that I've come ____ to quitting my job ]
1c. [ the closest *which I've come ____ to quitting my job ] ← ungrammatical
We can compare this example to the following non-relative sentence:
2a. I've come this close to quitting my job.
This should help illustrate the position of the gap.
The relativized examples 1a and 1b are functionally similar to noun phrases (although The Cambridge Grammar of the English Language on p.1169 says "there is no plausible reason to propose that the [...] phrases they head are NPs"). They can be, for example, the subject of a sentence.
Let's illustrate this by putting 1a into a complete sentence:
3a. [ The closest that I've come ____ to quitting my job ] is when my boss told me I had to wear that rubber chicken suit.
The optional presence of that and the correspondence to the non-relative 2a are both arguments against the alternative analysis you give at the bottom of your question. You were right the first time when you called it a relative construction.