For the purposes of this post, the Tube and other metro services are excluded.
Stations which come to mind are Canada Water and Rotherhithe, with around 550 metres between the two stations, or Bromley North and Sundridge Park, with about 700 metres.
Any more?
It isn't a well defined question in 2 respects.
If Metro services are excluded, how is Rotherhithe-Canada Water not part of that exclusion?
How short is short? And, given that Network Rail still measure in miles and chains, measuring in metres isn't very helpful.
On Thameslink, Blackfriars-City Thameslink is 28 chains (about 560 metres) and City Thameslink-Farringdon is 32 chains (about 640 metres).
On the Liverpool Street-Enfield Town route there are 13 intermediate stations in less than 11 miles. Of the adjacent station pairs, only Liverpool Street-Bethnal Green and Edmonton Green-Bush Hill Park are more than a mile apart. I think the closest are London Fields and Hackney Downs at 43 chains (about 860 metres).
Winchmore Hill to Grange Park on the Hertford Loop, visible from eachother.
I have this as 52 chains. Between Finsbury Park and Gordon Hill only Palmers Green-Winchmore Hill is more than 1 mile.
Anglia has various stations less than a mile apart out of London including:
37 ch Harwich Town-Dovercourt
52 ch Southend Victoria-Prittlewell
74 ch Colchester Town-Hythe
77 ch Brundall Gardens-Brundall