The good news is that it's possible to do it. The bad news is that vinyl flooring that old, and sometimes the adhesive, will often contain significant amounts of asbestos, and tearing it up is a sure way to contaminate a house and compromise your family's health.
You need to find a qualified analytical lab that is certified to test for asbestos (your local, state, or federal polution control agency is likely to have a list of reliable labs.) Then you cut out a small patch of the vinyl (maybe 1" X 1") all the way down to the concrete, and including the adhesive, and have it analyzed. This should cost under $100.
If there is no asbestos, you can tear and scrape, and use an adhesive solvent to remove the adhesive. If there is asbestos, this is not a do-it-yourself project (I know, because I worked in asbestos regulation for 20 years, and the specialized equipment is too expensive for a one-time job, and you don't have the knowledge and the safety training to keep it safe). You will need to find a certified abatement company to do this. They will work in safety suits, will build an air-tight enclosure to screen off other parts of your home, use special fans and high efficiency particulate removal vacuums, and transport the hazardous debris to an approved landfill.

Before going through all the trouble of tearing up what might be a hazardous material on your floor, even though I was not aware that sheet vinyl ever contained asbestos, you may want to just pull back the corners or lift the threshold and take a look at the slab and see if it is not cracked in several locations. My last house was a slab and when I ripped up my flooring, the concrete had spider cracks in it everywhere, especially in locations next to carpeting where the tack strips were installed into the slab. My guess would be that the amount of time and effort it would take to patch, paint or concrete stain and clear coat the floor would be similar to just putting in a new floor. I put in solid hardwood over the slab, with a vapor barrier.

pppft, heck, I'll just tile over the whole mess...

Ohhh, I hope you are joking...

Tile over vinyl is NOT a good idea. Too much flex.

disneyrsh-
I had about the same situation as you...a concrete slab with a layer of 9"X9" vinyl tile covered by a layer of vinyl flooring, both of which likely contained asbestos. If I'd had a plywood floor, I'd just have nailed new underlayment over it and tiled. I procrastinated for years and finally decided to put a third layer of vinyl over it all. It worked pretty well, although that's a recipe for adhesion problems, and the installer only agreed to do it if I signed a waiver. As Oruboris mentions above, putting tile on vinyl is not a good idea.

The only other suggestion I have, besides testing and a professional removal (if needed), is to leave the old floor alone and investigate whether it would be possible to affix underlayment to the old floor with a combination of adhesive and explosive fasteners, although this could be more work than you want.

I know you're not getting the easy answer you wanted, but at least you know what's what before you get into it and have a mess on your hands.

35 years old? That would put it about 1970 - Anyone know when they stopped using materials with asbestos? I know the house we tore down was built in the 50's and definitely had it. It was in the adhesive. We had to go through the whole removal process. I have heard that there are DIY kinds of testing where you take the sample and send it off. You may find out there is no asbestos and then you can continue as planned.

I believe that most, if not all, flooring manufacturers stopped using asbestos fibres in the 1980's in response to an anticipated ban on use of asbestos in many products, which the Environmental Protection Agency started working on in the late 1970's. EPA invested over 10 years to put the ban in place. However, despite the fact that the serious health effects caused by asbestos are very well-established, political and legal forces triumphed and the asbestos ban was ultimately not put in place. It is still legal for manufacturers to put asbestos in their products, although, from a practical standpoint, many of them have found sustitutes because of their fear of class action lawsuits. If you want to read the sorry history of this issue, here's a well-written, short article.

Here is a link that might be useful: Asbestos still not banned

I'll probably send in a chip to the lab to see if it has asbestos, if it does, I'll have it professionally removed, because otherwise when we sell the house I'd have to disclose it. Plus, we have 8 foot ceilings, and I'm not going to make the floor any thicker...

Hi disneyrsh,
Have your flooring issues been resolved? I own a professional concrete staining business in California, and I'd be happy to anwswer your questions. In the meantime, you may find some answers on our website, www.terrafina.us.

I have a question, id you dont mind terrafina. I am considering concrete in the kitchen and bath here in Phoenix. I have concrete floors under the tile. The concrete guy who came by said that he would lay something down on top to make it look smoother. Does this sound right?

jenny

Hi Jennye,
If you have ceramic tile on the floor, that will need to be removed, along with as much of the mortar as you can. It is usually impossible to remove the mortar completely, and you will also have "ghosting" on the concrete where the mortar was. You will probably be happier with an overlay. HOWEVER!! Please make sure that your concrete guy knows how to do that properly. Many people don't do it right, and then the material will begin to pop off, sometimes within months. Ask your contractor what materials he intends to use, and do some research on them yourself. The cement mix should have polymers in it, or there should be a polymer that is to be mixed with it. Polymers can be put on the concrete before the cement mix for added bond strength. Overlay materials stain different than concrete slabs: they take stain well (can get very dark), and may not mottle well. Be sure to get some samples on boards from your contractor. I hope this helps.

terrafina

Thank you so much for your answer. Yes, I have ceramic tile in the kitchen and baths. Do you think concrete will be okay in the bathroom with water? I am looking at trying to tie the whole house together with dark non-acid stained concrete in the kitchen and baths and wood everywhere else. It is a contemporary modern house.

Thanks,

jenny