I can't see the damage well enough to tell much about condition.
I'm not sure what you want to know so I can only assume that you have little or no tools nor experience woodworking.
The veneer can be removed rather easily but install new calls for moderate level of skill. The wood under the veneer is not fit for refinishing and once the veneer is gone,all othe surfaces will stand proud which is unexceptical.
If I understand you and the table,buying a piece of veneer,cutting out bad spots then patching the veneer,albeit imperfect,is probably the easiest diy solution. If it has no special meaning or value,replacement would be far cheaper than have it professionaly restored. Beond that is just a question of wherther it go's back to Louis XIV or Ikea on the 1st.
I

I was actually wondering whether I could just leave the wood without any veneer, i.e. a "rustic" exposed finish. It looks like a pine or some other cheap wood. In other words: What would the top look like without the veneer?

You'll never know until you remove it. Though--if you look at it from the under side, what does it look like? It might look like that on the top--except that it'll have glue, etc., and you might need to plane/scrape/sand to get a level surface. That seems like a lot of work, frankly.

I wondered if you could simply cut away only the bubbles, install some sort of leveling agent/putty/filler so you'd have a solid and smooth surface, and get laminate applied to the top. (a laminate shop could probably do it)

Some people really hate laminate, but I used to have a laminate-topped table (w/ a relatively decent wood look to it), and it was GREAT! I could roll dough directly on it, use cookie cutters, wipe up messes with a wet sponge, etc.
Seriously, I loved it enough that I have fantasies now and then of getting a similar table for the future. The one I had looked quite nice--it didn't scream "laminate!" (It helped that it was a dark woodgrain, and tight. And its texture was somewhere between glossy smooth and the pebbled feel that lots of Formica has.)

Or, reveneer the whole space.

(though veneer patching might be the fastest look--it would probably show a little.

Thanks. I probably could try to fix the veneer again. It's slightly lifting at the seams, so not terribly noticeable for most people.

Ok,here's what I would "realy" do in your case if wanting a truly one of a kind I could be proud of.
You can buy an inexpensive "inlay kit" that fits on a router. Rockler or any woodworking supply has them. Simple to use and a barrel of fun,not to mention creating interesting projects. You first chose the material for the inlay,in your case thin walnut or oak about 1/16th to 1/4th thick. Regular sheet veneer can be used but is often fragil to handle so I do'nt reccomend veneer until you have experience. Next deside on any desighn you can draw or trace from pictures as long as it isn't too small for you cutter define. There are also madalions and such in craft stores. I once used keepsake coins and military medals in a table given by a grandfather as a wedding gift. The trickest part is adjusting cutter leingth to exactly same as thickness of inlay. Adjusting cutter a hair short then sanding inlay after install works good also.
Don't own a router? Buy one,you only need a cheap one unless you plan to use heavy profile bits later on. Don't even think of asking me how to use a router.
Did I mention how much fun it is?

You can remove old veneer with a warm iron, or you can sand it off.

To fix a bubble, use a Exacto blade to cut a slit through the bubble along the grain and inject glue under the bubble. Roll it out with a heavy rolling pin over a non-stick mat (a marble pastry pin and a silpat backing mat work well). Weight it down while the glue dries.

If there is debris - like old glue granules - in the bubbled area, make a triangular flap, dampen the flap to soften the wood and lift it up. Clean out the crud and glue as described above.

If you have a missing chunk of veneer, lay the new veneer over the area and trim the edges through BOTH pieces. Then coax the old veneer out and lay the new stuff down.

Actually, what I have is not bubbles or missing veneer but lifting at the seams. I tried glue bu tit still lifts, probably because it looks like the veneer expanded a bit and overlaps a the seams.

Just not sure how much work it is to remove all the veneer and possible glue, and what the top will look like.

Did they use particle board in the 1920s?

Nnosoccermom wrote "Just not sure how much work it is to remove all the veneer and possible glue, and what the top will look like. "

Veneer removal albeit labor intensive (see lazygardens' suggestion) is possible and glue is even worse. The results after staining will be less than mediorce at best. Time better spent on something with better potential.

"Did they use particle board in the 1920s?"

No. It wasn't invented until decades later.

You repeat the same questions after hearing our responses,so in the interest of making some progress which seems to be eluding us , may I ask?
Are you at all interested in alternates to removing the veneer?
What are your minium expectations as far as appearance after going through the process? Are you optomistic the table will look better after removing veneer than it does now?
Other than breadboards on each end,are those lighter colored sections midway across the top solid or veneer?
Are you concurned about the fact solid wood sections will be higher than those where veneer was removed? What if anything will you do about that?
The "only" value I see resulting from removal of the veneer is education. The reward might be great for some if they sucessfuly repalced or repaired the damage where others might make a half hearted effort and count it as just another failure.
There is no silver bullet if that is what you are looking for. The result will only equal effort.

Thanks for your patience. You hit the nail on the head: I'm curious to see what happens when I remove the veneer, and I'm not terribly happy with the table as is. However, it also sounds that this may not be the project to satisfy my curiosity.
So, I will try to glue down the veneer where it lifts on the seams and try removing veneer on another, smaller piece. The light colored sections are solid wood all around [Somehow I had this vision of removing the veneer and having a gorgeous rustic table underneath. Sounds like that's not going to happen.]

I assure you the wood underneath would look realy bad next to the oak.

I believe you :)