If you're visiting Estonia and want to have been only just above Russian soil while travelling on land just for the sake of it then the Saatse boot might be your best option. There's a road from Värska to Saatse that crosses through Russian territory (entirely unremarkable forest) for a few hundred metres. You have to stay in a motorised vehicle and you're not allowed to stop (or travel below a certain speed), but it's surely the safest and most reasonable way of "visiting" Russia.
Streeview link
Even then, you shouldn't consider it risk free. I've been to Estonia 29 times and have seen most parts of it, but actually ATM I'd think it's overcautious but not unreasonably so to stay away from the border completely. The Russians ocassionally kidnapped people [who were high value pawns] across the border from Estonia even before more recent events, and imprisoned them as spies.
If you did want to do it, bus 94A leaves Polva at 0855, Polva railway station at 0904, Värska at 1020 and arrives at Saatse Museum at 1055. On the way back bus 94B leaves Saatse at 1200. (The practical effect of this is that it's not possible to get from Tallinn to Saatse via the Saatse boot by public transport in one day.)
Even in before times Russia wasn't actually too keen on people crossing that bit of the border as tourists; I tried to get Intourist to organise me a trip to Pechory and Pskov in the mid 2000s and they just told me to visit Moscow and St Petersburg!