This is definitely possible, even with smaller hands, though the execution might be tricky depending on the speed of the passage.
Play the first Db in the middle line and the Bb whole note above with the 1st and 3rd finger of your right hand while playing the low Gb with the 5th finger of your left. (ignore the top Bb, as it is tied to the note in the previous measure and should not be played again)
Jump your left hand up to play the second note of the middle line (C) with the 1st finger of your left hand (holding the low Gb with the sustain pedal). At the same time, replace the right 3rd finger on Bb with the right 1st finger (keeping the note held down).
Play the next note in the middle line with the left 2nd finger (crossing over the 1st finger) and the top Gb with the right 3rd or 4th. Figure out the remaining fingering from there.
Hope that made some sense! Swapping fingers on a key without releasing it will take some practice. It might help to practice each hand separately for a bit.
In response to your update, the solution is simple: just play the bottom Gb with the 1st toe of your left foot.
Joking aside (but seriously, that would be totally awesome!), I do not see any way to play this precisely as written, given that the top note is not tied. That said, piano scores with impossible intervals are not unheard of and the standard technique is, in fact, to play the bottom note just before the rest, catching it with the pedal (as others have suggested). With some practice this can be done quickly enough to almost be imperceptible.
Below is an example from Schubert's Impromptu Op. 90, No 3 of how this is often notated. (Fun fact: In the original score, Schubert just left it as is and let the pianist try to figure it out. The rolled chord notation was added in later by editors since it was impossible to play as written for most pianists. Go figure!).