A sorted S1 should be a joy in the twisties. But there are caveats...
The good news is the main tub is incredibly strong and can withstand major impacts. Unlike most cars, much of the chassis in an Elise is very easy to see, so look for ripples in the surface that indicate damage or poor repairs.
The rear subframe is the only significant steel component in the Elise and it does rot. It's easy to replace, but the subframe alone is £500 to buy plus labour to fit it, and this means swapping the engine on to the new subframe.
A more likely problem will be damaged suspension mounts. These are glued on with extra rivets to hold them in place in an accident. If they are damaged, they cannot simply be reattached or welded. Unless a professional repair has been affected, with evidence to prove it, walk away and find another Elise.
Luckily, these show up as an easily identified clonking noise and are straightforward to replace. The front suspension's ball joints wear too, so budget for replacing these at some point during ownership. Very early Elise S1s suffered from wheel bearing wear, but most should have been sorted by now.
Any Elise that doesn't track in a straight line should send alarm bells ringing. The steering rack offers a quick but not nervy unassisted 2.7 turns from lock to lock. If the rack is worn, it's expensive and involved to replace.
The early metal matrix composite (MMC) brake discs of early Elise S1s last very well, but are less well suited to track use as they don't dissipate heat as well as the later 282mm vented steel discs used all round. A set of MMC discs will be very expensive to replace, if you can find a set for sale.
Standard Elise S1s have five-spoke alloy wheels, with 185/55 VR15 front tyres and 205/50 VR16 rears. The same front tyre size is used on all models, but the Sport and 111S versions gained 225/45 VR16 rear tyres. The 111S has a 12mm wider track than the standard car's 1453mm to fit the wider tyres and the 111S also came with a new style of six-spoke alloy wheels.