Is there an opening sequence for black that is equally solid versus d4 and e4? Obviously you have to take into account white's subsequent moves, but the goal would be to have an opening where the first, say five or six or so moves (maybe with variatios in move order as needed), would be solid against most common e4 and d4 openings.
You could try Owen's defense, which is very much under-represented in all levels off chess in my opinion. Smirnov's 2-part series is nice and clear (although sometimes slightly over-simplified maybe), and he has another 2-part series about using this strategic system with the white pieces.
I would recommend playing the Colle-Zukertort at white a bit, the ideas are definitely useful for the b6/b3 systems. This also provides a very comfortable and active setup to shoot for as black when your opponent does not put a pawn on e4. The themes from the Breaking Stereotypes series provide some nice complementary insights for these positions as well.
Another option is the Sniper, which is a lot of fun to play and is maybe even more systematic as far as strategical planning is concerned (get king's bishop to the longest diagonal by playing 1. .. g6 2. .. Bg7, then play for central and queen-side activity via the dark squares by playing 3. .. c5) but it's significantly less forced which means the total amount of relevant theory is a lot larger. Also you might end up in some fairly theoretical positions, or positions that are really though to play unprepared. Super flexible though.
And then there's the Hedgehog, which is by far the most systematic and repeatable chess opening I know of. It's not uncommon to play the same 10 to 13 opening moves five times in row or even more, without your opponent even following theoretical variations.
(btw John Bartholomew is excellent at teaching chess through actual understanding so that's an unrelated recommendation)
Another possibility might be the Old Indian where Black plays d6, Nf6, c6 and aims for an early e5. You can also play like this against 1. c4. I think Nigel Davis did a Chessbase DVD on this, called "A busy person’s opening system". But I don't know it and I don't want to advertise it.
Although this might be "sound" more or less, I can't recommend such an approach. Not only will you fail to play the most challenging defenses against specific set-ups (as suggested in the comments) but also you might get stuck, playing the same (inferior) positions all the time. Chess is a rich game and exploring various types of positions (if only by playing them over the board) will not only improve your overall skill but will also enhance your enjoyment. And you will have much more to talk about.