Your best option to train for an obstacle race is to run obstacle courses. :) Now, this does not necessarily mean replicating the target course in your backyard. Instead, take a walk around your neighborhood and look for items that might resemble what you're going for in terms of obstacles.
Playground equipment can be particularly good for this. Fixed park benches (rooted in concrete) are good for practicing vaults. Most neighborhoods have at least one waist-height wall. There are various sources for poles to go over and under, although unfortunately, many of them are either relatively flimsy or in high-traffic areas such as in front of super-markets. Once you have your obstacles, include them in a run. Try to maintain speed, navigate the obstacle, then get right back to running.
Since the obstacles are not likely all in one place, your best bet will be doing the same obstacle repeatedly. For example, let's say that you've found that your local park has a section of ground with a steep drop-off of about four feet and about 15 feet of open space on either side. Run the 50 feet, hop up the embankment, trying to go in a smooth motion from bottom to top, and run the other side. Pivot and go back the other way, hopping down the embankment and landing smoothly, and run the distance. Pivot and return. When you find yourself tiring on that obstacle, use the walk between that and the next obstacle to recover and to determine whether you're up to the next one (a period of just walking can do wonders to help you decide whether that twinge in your calf is just muscle soreness or something more serious).
I highly recommend looking into Parkour for navigating the obstacles. There are plentiful YouTube videos on how to correctly perform the movements, in particular the landings. Amos Rendao has what I think is one of the best rolling tutorials out there (fair warning, the intermediate roll tutorial video typically has, as its preview image, his demonstration on the back of a comely traceuse, not actually nudity but appearing similar on a quick glance on a work computer). For the rest of the movements, that's contentious, but I've enjoyed the LaFlair set on vaults. If you're curious about the actual science of it, Parkour Science provides all of the equations that you might ever want to demonstrate, for example, why rolling on impact works, or what the maximum distance one can precision jump is.