Here is my list of some places to TRAIL RUN in and within an hour of Indy:
* indicates a DINO venue (or past venue), info available at DINOseries.com
*Fort Harrison State Park - two loops totalling about 10k unpaved trail, plus 3 mile paved trail loop
*Eagle Creek Park - 10+ unpaved groomed miles on east side, 5(?) more primitive trail miles on west side
*Southwestway Park - 4-5 miles of "spiderweb" trails.
*Southeastway Park - 2 miles of trail, plus a 1- mile "once-paved" loop and plenty of grassy area
Holliday Park - 2-3 miles of trail, lots of stairs
Marott Park - a couple miles of trail in 2-3 different parts of the park. Holliday and Marott are connected in the Holliday Park Trail Run route.
* Town Run Trail Park - 7 mile loop, very busy with mountain bikes... running can be less enjoyable during busy times
*Westwood Park (New Castle) - 10 mile loop, challenging and scenic
*Washington Township Community Park (Avon) - 3-4 miles of trail ranging from easy to really steep
*Brown County State Park - stellar scenery, with hiking and MTB trails there is probably 20 miles
*Morgan Monroe State Forest - many excellent options, including 3 Lakes Trail, Tecumseh Trail, etc...
*Yellowwood State Forest - many trail options, site of Tecumseh Trail Marathon finish line and fun run (5 mile trail around the lake)...
Johnson County Park (Edinburgh) - you can run a rat-maze of grassy trails through this flat brushy woodland.
There are more, these are just the ones that came to mind.
Regarding safety... I would feel pretty safe (even as a woman) on most of these trails. Carry a cell phone in case you get hurt, and I suppose a pepperspray might add peace of mind.
I love trail running. I can run all day on trails... pavement tears me up. It is so much more interesting and a more well-rounded workout.