It can seem like there's a catch-22 when trying to break into product. You feel like you won't get an opportunity to do product if you don't have any product experience and you can't get any product experience if you aren't given an opportunity to do product.

But it only seems like a catch-22 if you make an incorrect assumption. The truth is: you do not need any product management experience to be hired as a product manager.

I'll say it again, because it's one of the hardest things to realize and it's also one of the most important. 

you.
do not.
need.
product management experience.
to be hired.
as a product manager.

Notice that I wrote "product management experience," not "experience doing product." It's subtle, but those are two very different things.

Product management experience just means you've had the PM title and official responsibilities. There's something to having been in the position and all the pressure and ambiguity that comes with it. But it's not the core of the job.

Experience doing product means you've actually focused on making something that customers want to use. And that is something you can do no matter what your official title is.

"Doing product" means you cared about making your customer's lives better through your work and you contributed meaningfully to your team turning an idea into reality. Product managers never work alone, so you don't have to be the "decider" to do product.

It means you understood you understood who your target customer is (maybe it's you?), you articulated their social, emotional, functional needs, and you created something that met those needs.

Have you done any of that? Then you've done product. When you're asked whether you have product experience, talk about that.

One of the best product manager definitions I've read is from Shreyas Doshi. Read through it but pay attention to what's missing: 

Even the key skills and traits are not product-specific. If you can demonstrate them in any role, that's a very good signal that you'll be good at product. If you want to focus on developing any skills, I recommend picking one of these that you're already pretty good at and get even better at it.

After all that, if you still feel like you haven't had an opportunity to practice those skills in your job, give yourself an opportunity. Your application process is a product. You're the customer. Make it work better. Talk about how you did it in the interview.

BOOM.

You're doing product.
 
Addendum: As Kelsey Hightower points out, you don’t even really need specific experience with the tools of the field that you’re being hired into. 

More important is the ability to see what those tools are and learning how to use them when you need them.