I have assumed that the elements of the string can be in any order and appear any number of times. For example, 'a foo' should match and 'a foo b foo' should not.
You can do that with a series of alternations employing lookaheads, one for each substring of interest, but it becomes a bit of a dog's breakfast when there are many strings to consider. Let's suppose you wanted to match zero or one "foo"'s and/or zero or one "a"'s. You could use the following regular expression:
^(?:(?!.*\bfoo\b)|(?=(?:(?!\bfoo\b).)*\bfoo\b(?!(.*\bfoo\b))))(?:(?!.*\ba\b)|(?=(?:(?!\ba\b).)*\ba\b(?!(.*\ba\b))))
Start your engine!
This matches, for example, 'foofoo', 'aa' and afooa. If they are not to be matched remove the word breaks (\b).
Notice that this expression begins by asserting the start of the string (^) followed by two positive lookaheads, one for 'foo' and one for 'a'. To also check for, say, 'c' one would tack on
(?:(?!.*\bc\b)|(?=(?:(?!\bc\b).)*\bc\b(?!(.*\bc\b))))
which is the same as
(?:(?!.*\ba\b)|(?=(?:(?!\ba\b).)*\ba\b(?!(.*\ba\b))))
with \ba\b changed to \bc\b.
It would be nice to be able to use back-references but I don't see how that could be done.
By hovering over the regular expression in the link an explanation is provided for each element of the expression. (If this is not clear I am referring to the cursor.)
Note that
(?!\bfoo\b).
matches a character provided it does not begin the word 'foo'. Therefore
(?:(?!\bfoo\b).)*
matches a substring that does not contain 'foo' and does not end with 'f' followed by 'oo'.
Would I advocate this approach in practice, as opposed to using simple string methods? Let me ponder that.