It seems to me to be a very solid structure that can play very good in the endgame.
Example: pawns on c2, c3, d2, d3.
It is a solid structure indeed, avoiding one of the most important weaknesses of doubled pawns, which is that they're generally hard to protect. Also, they're less easy to block than just a c- and d-pawn, since the backward pawns can attack enemy pieces on the blocking squares.
However, this structure only influences the b-, c-, d- and e-files, while a 'regular' connected pawn structure influences 6 files. Such a structure is also more likely to be able to force breakthroughs. Everything depends on context, as user1583209 notes, but in general it's better not to have doubled pawns.
The structure itself isn't weak, since as you mentioned all of the pawns are solid and protecting each other. The problem is that you can't do much offensively with these pawns. Take the following example:
Black has his c-, d-, e-, f- pawns.
White has the pawn structure you mentioned.
Notice how White is unable to block Black's f-pawn. Also, how is White supposed to do anything in the center? He can only really use his front 2 pawns in an attack, while the back 2 have to sit and wait.
It depends on the position though. Maybe you don't want to move any of your pawns, and controlling the squares around that 4-pawn block is more important.
The structure lacks mobility and is therefore inherently weak. The c2 and d2 pawns can not advance if attacked.