Just to address this in a generic way to begin with
...is it possible to sand, reapply finish, and then somehow smooth over the border between the "new" and "old" finish?
Yes this is possible.
A lot of sources do state it's impossible while not saying much about why, even those that do go some way towards an explanation tend to repeat well-worn platitudes like "it's impossible to hide the edges" or "a tide mark is inevitable".
Anyone who has ever done this successfully knows these are bunk. Feathering in can be done successfully.
What I think is true however is that you can't go into a job like this knowing you'll be successful, even with previous good outcomes under your belt.
Sometimes (rarely) this goes quite smoothly and actually seems easy, if it's your first time you wonder what the fuss is all about. But in some cases it's not as straightforward as in others and things just don't go well.
[is] the only way to make this "look good" is by refinishing the entire room?
Some sources will certainly tell you this (and not just those trying to get more business).
Re-doing the whole floor is undoubtedly a way to ensure the room ends up looking uniform. Whether it will then match the floor in the hallway or an adjacent room is another question......
If you have imperfections in the finish layer only
This is relatively easy to rectify. In fact it's actually routine to do this kind of thing, although it does require care and attention.
There are many guides on how to 'perfect' or polish a finish, which all involve some amount of levelling or flattening off (via scraping and sanding), followed either by further abrasion up to a very high grit1 or, careful application of a final coat or two over the levelled/smoothed existing finish2.
If you have imperfections in the wood
This is what's trickier to do, since you have to sand down to bare wood, sand it enough to remove the imperfections, and then apply new finish to the bare wood and on to the surrounding edges of the original finish, hope the colour ends up uniform, and then try to blend in the new finish with the existing finish. In addition to the general difficulties of feathering in3 there are two additional factors which present obstacles:
- the wood may have changed colour due to light exposure.
If so the freshly sanded area will be lighter (darker woods) or darker (lighter woods), and/or a slightly different hue, and/or may be more vibrant or saturated4. All of these possible colour differences may even out over time, but it can be a long wait with many species — a few years at least.
- the finish may have changed colour since it was first applied. So even if you have the exact same can of finish that was used originally it is no longer the same colour as the stuff previously applied.
There's no getting around this particular problem using only the original finishing product. While tinting the finish or using other colouring tricks are possible either is a shorter-term fix, since what will happen eventually is that the colour-matched new finish will subsequently end up darker than everything around it once it too darkens naturally.
1 As you sand to higher and higher grits at some point no discrete sanding scratches are visible, and eventually you will actually be polishing the surface, although generally to produce a true high-gloss surface polishing agents need to be employed at the end of the process.
2 In furniture work finished in varnish this is now often accomplished by one, possibly two, final dilute coats (i.e. wiping varnish). Applying thinned varnish and wiping away some or all excess makes it relatively straightforward to get a good result, with minimal or zero chance of any bad dust nibs.
3 Bonding issues, tide marks or 'witness marks'.
4 Pine darkens a lot and will go more burnt-orange or tan, oak will tend to get darker and go browner, walnut will lighten and tends to get more orangey.