I asked my career counselor about how to break the chicken or the egg scenario where a given job needs years of experience but how is one supposed to garner that experience without having had the job? Answer is internships paid or otherwise and only rich kids can get the unpaid internships so if you're lucky enough I vote go for it and milk it for all its worth. Building from nothing is quite a bit more challenging to say the least.
The career counselors at my university were complete idiots. I asked about the same chicken-or-egg problem you mentioned. They gave me some bullshit answer that didn't help at all.
My solution was a combination of tactics, although it may only work if you're in some kind of engineering field.
First, I taught myself practical, real-world skills that they don't teach in the classroom. With that, I started my own company. I worked cheaply at first, since I was just trying to get documented experience *and* I had some strict work constraints. Basically, I only took clients that would let me work from home after-hours and wasn't full time. Those clients are hard to find, especially while starting out. But one thing for sure is that companies would rather deal with a vendor (your company) than you on paper. Then, on your resume, you can say you worked at a company. Doing your own thing also gives you a huge advantage over people that go directly into big-corporate: you can complete an entire project from beginning to end, being involved in all aspects. Big-corporate projects are too big for one person, so you hardly get a chance to generalize.
So on top of contracting, I would keep writing research proposals. Eventually, I was brought in to some PhD level research as an undergrad. It was some pretty recognizable stuff. So I was never an intern; I was a researcher. Even though the pay was awful, it was still a "real" position on paper.
I got pissed at school and quit. Immediately, I contracted full time for a medical imaging company. I did some cool stuff there, but my boss was a total prick (you can search for my "wtf, my boss is on drugs or something" thread on ATOT) and I left. Immediately, I had companies swarming me and it wasn't a problem.
Biggest advice I can give you:
1. Make your OWN experience.
2. Ignore what your counselors tell you.
3. Talk to people IN INDUSTRY and get a feel for things.
4. Do not follow resume writing advice provided by your school - ask industry people to send you copies of stellar resumes and write your own.
5. Don't drink the Google, Amazon, or Apple punch. In other words, get a good job in a professional environment for high pay, not a "cool" job at a "cool" place because you're young and don't know any better. Money talks and bullshit walks.
Hope this helps.