Sorry to hear the job offer fell through, but good for you treating it as an opportunity.
If it was me, I would ignore the $10k in capital and only pursue a venture that I could get off the ground with almost no up-front costs (say, a couple of hundred dollars or less). I would only start to invest money in the venture once it was already profitable, and then I would only put in small amounts of money that I was confident would lead to greater profits.
If had enough income from other sources to cover my expenses as described, I would definitely only do something fun and that I could envision would remain fun even when I was doing it more and more, as a business. That part seems more difficult than it appears on the surface, because there have been a lot of things that I've thought, "That would be fun" that would quickly become a chore as soon as I had to do it on a regular basis. And if my goal was to launch a growing enterprise as described, I would make sure to only take on ventures where I could outsource parts of the business so that if it became very successful, I wouldn't be tied to 60 hour work-weeks keeping it that way. So something like dog walking would work, where once you've established a clientele and proven track record, then you could hire some reliable students to take on the majority of the work.
As to WHAT I would do, who knows! Since I'm expecting to ER in one month, I've put more thought into this lately. Right now topping my list is teaching martial arts/self defense/security/protection, probably in one-on-one sessions (not groups or classes) with high profile (i.e., wealthy) clients. I might also teach/coach in a skill I've acquired over the years that I'm not ready to divulge, in order to keep all you highly enterprising vultures away from it ;-)
I might also consider gaining expertise in some kind of niche market where I could resell items on eBay/etc, preferably in larger ticket items. I would have no interest in doing high volume, would prefer to focus on something where I could turn a profit of $100+ on any sale (I have no interest or experience in antiques, but that might be an example of somewhere you could gain expertise and get nice returns, or perhaps art work). That might be another case where you could outsource the grunt work (shipping, customer contact, etc.) and focus your expertise on locating quality items that would generate large profits.
And of course real estate investing, though capital is clearly required unless you can line up a financier.
