The Richard Shultz Petal table - Renovation.  

Again, great find!
I would keep the base as original as possible.
Maybe just a gentle cleaning?
You don't want to make a new table out of it, no?
Originally these were varnished, I think, back in the sixties? If you don't want to do this, I would suggest some kind of wax. It will give more body to the surface, where oil gets more absorbed and forms a thinner film.

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Epifanes Spar Varnish is top-shelf.
If you're going to use the table outdoors, spar varnish is probably the ultimate finish for long-term protection. Redwood, like teak, is naturally rot-resistant and will quickly turn a lovely fox gray left unfinished. The boards used on your table appear to be quarter-sawn (or, more accurately, "vertical grain"[VG] when refering to softwoods) and should remain quite stable in use outdoors or in. Old-growth VG is the finest grade of redwood. It is still relatively soft, however, and dents easily, so film finishes like varnish or paint can be compromised by scratches, dents, etc. and *must also be maintained*.
Regular maintenance is the key to outdoor wood finish performance. There is no miracle product. In some locations and for oil finishes, that can mean more than once a season.
If you do decide on varnish and don't want a high-gloss look, stick with the gloss product and rub the final coat out when fully cured with #0000 steel or bronze wool for a nice satin sheen.
Oil requires frequent application depending on exposure and tends to mildew on surfaces that don't get sun. I originally used it on some VG redwood deck chairs I made more than 20 years ago but then let it wear off and the set go gray, but I'd still recommend varnish for a tabletop.

My default answer
I like to use an oil or rubbing varnish simply for the ease of maintenance. Maintaining hard poly requires too much stripping. And spar varnish in my experience likes to crack and delaminate especially on soft woods.
But I think the better questions are: what was original? And where will you use the table? (Outdoors or in, high uv environment or Seattle).

OK
I am pretty sure it was varnished originally but had since been refinished w/ something foul that clouded quite a bit (and it was a messy job too boot). I had the option of just trying to take that off but elected to plane off the finish and even out the boards. I am pretty happy with that decision. Now just need to round the edges a bit.
One of the boards had come loose and was awkwardly glued in place. We got that off but now need to figure out how to reattach that.
as for painting the base, I never really thought that was going to be necessary. It has its issues but overall is in acceptable shape. I will go at it with some toothpaste and toothbrush when I get home.
At this point I am thinking an oil based product (Danish oil-ish). I am going to keep this indoors and if I move it on I would not be surprised if the eventual owner did as well.
So, kind audience, what are the best of brand Danish-oil-like products I should consider? I am REALLY interested to see how this looks once that has been applied.

as an aside
the guy who helped me with this makes the most beautiful surfboards in the world, vaneered w/ any number of Oregon hardwoods. If anyone is interested in seeing them let me know (I own one and don't surf, I traded him a wonderful danish dining table for it. I consider it art).