Whee, someone posts a retro problem. This is quite messy; I just put all my thoughts as I solve this problem, so there might be unnecessary/unclear steps.
Black made 6 captures (c7xd6xe5, d7xe6xf5, g7xf6, h7xg6). This is exactly everything that White is missing: four pawns, one rook, and the light-squared bishop. (White's pawns might have promoted to replace some of the existing units.) All captures happen on the d-g files, but White's missing pawns are on a,b,c,h files, so they either captured or promoted to get to the d-g files (or to replace some original units existing at the moment).
Black is missing the two bishops. With only two units, White's a-pawn cannot reach d-g files, so it must have promoted. But there's Black's a-pawn on a7-a6 (Black didn't make any more captures, so it's impossible to have the cross captures a7xb6 and b7xa6). Thus White's a-pawn must have captured at least one unit to move out of its file. (One unit is enough, if it captured a6xb7.)
Similarly, now that we account one of Black's missing units, there is only one more capture left, so White's b-pawn cannot reach d-g files either. So it promoted, capturing the remaining dark-squared bishop to reach c-file and promote. White's c-pawn and h-pawn walked straight to promotion squares, and White didn't make any other capture. Also, indeed White made Pa6xb7, the only way for the a-pawn to promote with only one capture.
Now, what was the last move? White's pieces besides the king are blocked. White's pawns are also blocked; h3xg4 is impossible since White's captures are all accounted for. Kf4-g3 is impossible since the king is under two contact checks (Pe5 and Ng2). Kh3-g3 is impossible; Black's Rh2 is trapped (no way to not give check before it). Thus Black made the last move.
Black's last move was clearly not by the rook or the king. Nh4 is blocked, while Nf4-g2 is impossible as Black would be giving check before his turn. (Nf4xg2 is impossible because Black cannot capture any more; all captures are by pawns.) Likewise, Rh3-h2 is also impossible. So Black's last move is with one of the pawns.
e7 and f7 didn't move. b7-b6 is impossible; White needs to unpromote the a-pawn first, which requires b7. Similarly, h7xg6 is also impossible; White needs to unpromote the h-pawn which requires h7. g7xf6 is impossible; this blocks Black's dark-squared bishop on f8, while we need it to complete White's captures. We have three candidates: a7-a6, d6xe5, or e6xf5. The former simply passes the last move to White, which we have proved impossible above (a7-a6 doesn't do anything to White's completely locked position), so the last move was d6xe5 or e6xf5.
How do we extract the southeast cage, with all of White's units and several of Black's pieces? Since White's units are locked, the only way is to retract some of Black's pieces. Which one is it? The only two units that have prospects of leaving first are Ng2 and Nh4. The former is impossible, however, for the same reason why Nf4-g2 is impossible as a last move (Qg1 giving check). So Nh4 must be extracted first.
Ng2-h4 or Nf3-h4 is clearly impossible (f3 and g2 are occupied, and they weren't the first ones extracted). Nf5-h4 would be giving check before Black's turn. So Ng6-h4 it is, which requires retracting h7xg6 first. But as above, h7xg6 cannot be retracted until White unpromotes his h-pawn.
Where's the h-pawn, though? We need to retract one of Black's captures, which should be led to h8 as fast as possible so it can retract to h7 and then h6.
How many moves do Black have until h7xg6 can be retracted? There are only four: a7-a6, d6xe5, e6xf5, and d7xe6. (c7xd6 is impossible since c7 is required for White's b-pawn and c-pawn; the others are impossible for the same reason they cannot be last moves.) Once we retract Black's move, White has only four moves to complete his task. At first, this seems possible: retract -1. e6xRf5 Rh5-f5 -2. d7xe6 Rh8-h5 -3. d6xe5 h7-h8=R+ -4. a7-a6 h6-h7 -5. h7xg6...
...but alas, this approach fails short; White checks Black after White promotes (move -3)! So we need to make a shield, for example -1. d7xNe6 Nf8-e6. But now we're only left with three moves for White! Two are for h6-h7 and h7-h8, so White must move, in a single move, to one of the capture squares (e6, e5, f5). This is impossible. (The only one that looks possible is e5, but Black's f6 pawn is blocking.)
Thus the answer is no, Black cannot castle.
A famous puzzle was introduced by Nikita Plaksin in the following diagram and the question is "Can we reach this position without losing the long castel for black?"
Please provide a game to illustrate your answer.