The basic meaning of なし is the same as ない, but its grammar is different.
Long ago, adjectives used to have separate sentence-final forms (with し) and pre-noun forms (with き). People gradually started using the き form at the end of sentences too, and eventually the /k/ dropped out, turning it into い:
赤し sentence-final form (historical)
赤き pre-noun form (historical)
赤い sentence-final form AND pre-noun form (modern)
A few adjectives had the し ending built-in as part of the root. They conjugated pretty much the same way, except the sentence-final form didn't need an extra し added since one was already built-in:
美し sentence-final form (historical)
美しき pre-noun form (historical)
美しい sentence-final form AND pre-noun form (modern)
The し ending is now lost for most adjectives, although you'll hear it sometimes when people quote the literary language. The き ending too is considered literary, but you'll probably recognize it from literary contexts such as song titles, where it's used to this day.
However, certain adjectives have fossilized forms available that are still used in modern Japanese as modern words, without the feeling that they're literary forms. なし is one of these. Etymologically, it's the same word as ない, but it should be distinguished in modern Japanese because its usage is different.
Of course, なし can be used more than one way, and some of those ways overlap with ない, which is unsurprising since they're historically the same word. Let's focus on the way it's used in 問題なし:
It attaches directly to nouns without giving the impression that anything is omitted:
Although phrases like 問題ない are relatively common as well, I think 問題がなし is significantly less common than 問題がない. It's probably best not to consider 問題なし as a case of ellipsis.
なし is commonly treated as though it's nominalized, in which case it can be followed with the copula だ, genitive の, adverbial に, etc.:
In this sense, we can say なし means ないこと. In this usage, it's often characterized as suffix-like or as forming compounds, but it can be treated as nominalized even when it's not attached to a noun, so "suffix" doesn't work as a complete description.
It can be translated "no 〜" or "without 〜" in many cases.
問題無し (literally "no problem[s]" / "without a problem")
Although なし can be treated as nominal, it doesn't have to be. You can use 「問題なし」 or 「異常なし」 and such as complete utterances, if you like. But any large dictionary should have plenty of examples where it's followed by particles that follow nominals such as だ・の・に・で・と. Let's take a look at some examples from 研究社 新和英大辞典:
work without even a holiday
an aspirin-free, sugar-free tablet
Relax. Stop being so formal.
If you take the day off without even telling me, it causes problems.
One would be hard put to deny that things have taken a step backward.
Let's not have any secrets between us.
There are plenty more! It's common to use なし this way, even though it was historically a sentence-ending form. One more usage of note is as a conjunctive form, similar to なく. Here's an example from Martin's 1975 Reference Grammar of Japanese, p.833:
Even now, you know, at my house there's no radio, there's just busted black-and-white television.
Personally, I suggest you memorize なし as having its own set of grammatical patterns in modern Japanese, and pay attention to how it's used. You'll find that certain phrases with なし are especially common, and your example of 問題なし is one of those.
(There's probably more to say about it, so watch for additional answers!)