Is it possible to keep live plants in a fresh water aquarium without bottom gravel?

There are ways. You could choose plants that don't root into substrate - like floating plants or plants with rhizomes. Or you could get terra-cotta pots, which are safe for aquariums, put some aquarium-safe substrate in them, and put aquarium-safe rocks on top so the fish can't easily uproot the plants.

However, that you have Goldfish makes things more complicated, because they love eating plants. Java Fern is an option, as it tastes so bitter that Goldfish tend to leave it alone. This plant grows with a rhizome. That means it would actually die if buried. Instead, you tie it to a piece of decor with cotton thread - a stone, wood, etc. or leave it untied in the tank. Anubias is another option, as it has very tough leaves. Like Java Fern, it is a rhizome plant. There are a variety of types of Anubias that look different, and if you search them out you can get a fair variety of sizes and leaf shapes with Anubias. Java Fern and Anubias are also easy to grow, tolerating normal aquarium lighting and not needing fertilizer beyond what is produced by the fish. You basically just need to remove any dead leaves.

No. without some type of substrate, plants float around the tank and die

Yes and no. Some plants can float on the surface. Others need a gravel to root or just keep them down. But many fish will tear up the leaves and fouling the tank a lot faster.

Some plants yes, others no.

But. ..... goldfish will eat or tear up most plants for fun

And. ....live plants require better lighting than what comes with most cheap aquariums. Regular lighting just grows algae well, not plants.