Is it realistic to completely avoid PvP in EVE Online?

EvE is a purely PvP game with minimal PvE elements mixed in for money making. It is possible to sit in a station and just trade all day, or do planetary interaction or other extremely dull aspects of the game - but the game mostly revolves around spaceships fighting other spaceships.

There are missions with NPC pirates and space-rocks to fight (mine), however the missions aren't very deep and really there isn't enough content in PvE to enjoy. The game is PvP-oriented in that your internet space ship should be fighting other player's internet spaceships.

Think of it like chess. You can move your pieces around the board for no reason, but ultimately to play the game you need to attack your opponent. It just doesn't make sense to not play competitively.

edit: I would like to add that, there is some interesting PvE content such as group mining, corporation management and wormhole exploration - but all of these come with some PvP aspect to it.

I played EVE for about 2-3 years, not counting a 6-month hiatus in the middle when I didn't have the time to play.

Contrary to what almost everyone else is playing, I can assure you that avoiding PvP is most certainly possible, about 99% of the time. The key is to play smart, and always be vigilant -- someone can attack you at any time, in any place.

• Choose a low-traffic/low-population system to use as your home base. Amarr space is full of clusters of high-sec systems with almost nobody there -- I spent most of my time in the game mining there, even though my character is Gallente. You can fly where ever you want to in EVE. Low-traffic systems also happen to be more profitable for carebear pilots, as there's less competition.
• Be vigilant. Keep your eye on Local and on your overview, and dock up if there's a sudden spike in the local traffic or if someone suddenly appears near you and approaches.
• Always have an escape route planned, no matter what you're doing. Learn about safe spots and alignment -- Google "EVE safe spots" to learn more, as going into the whats, whys, and hows are well beyond the scope of this question and this answer.
• When in doubt, "safe" (i.e. go to a safe spot).
• Always have multiple safe spots in any system you frequent. Never stay in one long, and always assume that any "safe spot" where you leave behind stuff (a can, a spare ship, etc.) or where you sit for more than 30-45 seconds is known to others, and remain vigilant.
• When you have to visit high-traffic systems (e.g. Jita), don't dink around -- get in, get out. Suicide gankers abound there, but there's so many juicy targets that the odds of you getting ganked are low -- but rise rapidly the longer you linger!
• When you have to visit a low-sec system, get in, get out. Don't dink around. Assume that everyone in Local will attack you -- and slaughter you! -- on sight, so don't let them sight you.
• When you have to visit a null-sec system, assume that you will not leave alive. I suggest you don't even bother -- you won't leave alive. (That said, I did survive my one-and-only trip to a null-sec system, but only because it and the adjoining low-sec system I entered through were completely empty.)
• The safest "carebear" profession in EVE is exploration. This is because you will spend 99% of your time at safe spots and at exploration sites, making you harder to find. Exploration in low-traffic systems is almost completely safe -- even when Local lights up with a dozen pirates, you can be assured that you will most likely not be found, at least not before you're able to dock up at a station. NEVER probe from anywhere except a safe spot, and NEVER sit still -- arrive at your safe, hit the engines to full speed, then launch your probes and start scanning down sites. Periodically hop to other safes (unless Local is completely empty). Bookmark the sites you find -- those are new safes to add to your collection after you harvest them! Unfortunately, exploration is not easy to get into for new pilots.

If you fly smart, you will avoid PvP almost entirely. I've suffered only 3 PvP losses (no wins, I'm not a PvP pilot at all) in almost 3 years of flying by just being smart about where, when, and how I flew; none of my losses were crippling, because I followed the Number 1 Cardinal Rule of EVE: NEVER fly anything you can't afford to lose!

Fly safe!

Edit:

Almost forgot the biggest error of new pilots: Don't autopilot. Sure, it might seem convenient to set up that 40-system hop, hit the autopilot button, and then walk away from you computer while your ship does the flying for you, but more often than not when you come back you'll be staring at your pod floating in a field of debris, or even worse you'll be in a fresh clone back in station! Set your route, then warp-to-zero to each gate in your route and jump immediately.

That brings up another tip: Warp-to-zero to gates and stations ONLY. When going to belts, moons, etc., warp to 100km, scout it out, then warp away before warping back in to a closer point if it's safe. More than once I saved a very expensive Hulk from a ganker lying in wait in a belt by doing this -- even a Hulk can align and warp away before a ganker can close from that far away and lock you down.

As someone who doesn't do PvP, i would like to disagree with everyone else and say it is possible.

Yes staying in high sec leaves you with the chance of being suicide ganked or can flipped.

But in 4 months of eve (which isn't long true) i have never been suicide ganked. Joining a corporation helps prevent some, but be mindful if you ever get into a corp war.

Can flipping is common but follows one easy rule. if they steal from you, never take it back, never shoot at them (it could be a lure, where the flipper has a stealth-ed buddy waiting to take out what you take him out with). if your can mining, you get rid of the hassle of moving back to the station every time your cargo is full for security.

Since you can get a trial and your absolute sure you don't want to be part of the PvP system, you could use that to see if the missions or mining could keep you interested in the game or if you just like the freedom of flying your spaceship.