Most of my travel has been as a single mom, with my child. However, during international trips, he has sometimes stayed in the room (wiped out!) while I was still wide awake and ready to explore.
Generally, I've felt pretty safe wandering around a foreign city at night by myself. I've done a 10 pm to 2 am walk around Bruges at night by myself. Never once felt unsafe-- it's a city with a lot of night life and a lot of people around.
I've also walked around Berlin, Edinburgh, Paris, and other large cities late at night, alone. I feel more safe walking those cities than one close to home-- Portland, OR.
I did my first 5 day long, solo road trip from Oregon to Utah this summer. I slept in my car (Subaru Forester) for two of the 4 nights. Once in a campsite in the middle of nowhere (next to Swan Falls Dam, in Idaho), and another in a truck stop parking lot. I covered my windows with cardboard I had cut to fit. No one walking buy would be able to tell who was in the car, which is partly why I did it (the other reason being I like it completely dark when I sleep).
I also carry my gun (Utah permit allows you to carry in multiple states).
I'm against getting a "large dog for protection". You are a dog lover and want a dog, or you don't. A large dog may or may not offer you the protection you are looking for, but it's not fair to the dog if that's the only reason you got one. Just my opinion.
I feel like I have a healthy fear of my surroundings without being paranoid, but I know I am way more comfortable in solo situations than many people are. And I often scare my father! (He's a retired sheriff...) He never took issue with making me walk home from work near midnight as a teen (he wouldn't let me get my license, but then, 5 years later, he and my mom drove my brother the same route during daylight hours because it was "too dangerous" for him to walk, until he finally got his license at 21). I think my dad, being overprotective in many other areas, but not over this (for me anyway), has helped me to "not care". Of course, now that I "don't care", like he didn't, he's concerned...