The problem is not only "tall" vs. "high". "X with Y" requires two nouns or noun phrases, but "tall" and "high" are adjectives.
That being said - as @Ben McGah has pointed out - "X stories tall" doesn't sound quite natural. "X stories high" would be better. As he also pointed out, in American English we write "story / stories", while in British English it's "storey / storeys". You can avoid the ambiguity by using "floors" instead.
The dome is 710 feet in diameter and 18 stories high
The dome has a diameter of 710 feet and a height of 18 stories
The dome's diameter is 710 feet and its height is 18 stories
By using "-story", you can imply "height" without needing to spell it out:
- The 18-story dome is 710 feet in diameter
You could do the same for diameter:
- The dome is 18 stories high and 710 feet across
Don't overdo it, though; I would consider this combination to be over-simplified to the point of diminished clarity:
- The 18-story dome is 710 feet across.