Yes, I have applied for jobs out of my league, and even managed to get them. 🙂 When I was 23, with 1 year as an accountant at a super-small tax and bookkeeping firm and 1 year as an accounting clerk in a large company, I applied for a Controller job. It didn't have a specific experience requirements, so I thought WTH, might as well try. I got the job. (I was 24 by the time I started…but sounds better to say I was 23 when I applied haha.) I had people tell me that there was no way I could do it, I wasn't old enough, didn't have enough experience, etc., but 18+ months into the job, I've excelled at it. It's a small accounting office, so as Controller I do some things that usually would be Staff Accountant or even Accounting Clerk, but I also do all the things that would be Controller.
I've also applied for jobs out of my league that I didn't get, but I got through the nerves of my first accounting interview by having it be one that I didn't qualify for (though granted, I got the interview through cold-calling and didn't know what they were looking for or whether I qualified till the interview), which helped me be more comfortable for the first one that I had a shot at (and which I landed).
All that being said…there are some that I don't think are worth the time and effort to apply to, unless you're out of a job and desperate. If I was in your spot, employed with 5 months experience, I probably wouldn't apply to one that required 5 years' experience. However, if I was unemployed with 5 months experience, I'd probably apply to it. So, I guess I wouldn't see it as impossible, just too improbable to be worth my time, if I already had a job and wasn't in dire straits to switch. But, I'd apply to something that required 5 years experience if I had 3-4, or apply to something that required 3 if I had about 2, etc.
One caution: if you're ever looking at governmental jobs, some of those have specifications in the application that by applying you swear that you meet the stated minimum qualifications, so for those, only apply if you have the experience it states. I applied to one state job that had such a statement in its application – thankfully that was one I did truly meet. 🙂