TLDR: Yes, in general it's perfectly fine to use an indefinite article before a group of nouns.
But why? And what exactly does it mean? That's a great question. There are two ways to parse such a construct.
One that is suggested is that the indefinite article applies only to the first noun. In that case the interpretation of "I am going to town to get a burger and chips" would be that you are going to get two things: 1) a burger and 2) chips. However in the second case that interpretation for "They do a nice burger and chips" would have to mean that they do 1) a nice burger and 2) chips. That's almost certainly not what is meant in this case. The chips are meant to be nice as well.
What is probably meant in this case is that "burger and chips" is a noun phrase. The noun phrase can take an indefinite - or indeed definite - article. For example "I had the chicken sandwich, you had the burger and chips".
There is ambiguity here. "Nice burger and chips" could mean that only the burger was nice. Context usually gives the meaning. If the pair of nouns naturally belong together then they are likely to be a noun phrase. If not then they are taken separately. "I had a nice cheesecake and salad" probably means that only the cheesecake was nice. If you meant this in the original case, saying "I had a nice burger and some chips" would make it clear.
In any case there is nothing ungrammatical about the usage. It's just one of those cases where the meaning is technically ambiguous and determined by context and usage.
NOTE: For our North American friends "Burger and chips" in British English is what you would call "burger and fries", and they do belong together in a noun phrase.