Cat5e can handle 10GbE over short distances with proper termination. Spec'd to, what? Around 35-45m? Cat6 is spec'd to I think 55m and Cat6a is spec'd to the full 100m for 10GbE. I have seen people run 10GbE over Cat5e over modest distances. I don't think I've ever seen anyone try to push the limits, but I've seen 15ft connections over cat5e and 10GbE between a server and a switch that hit full specification on link rate AND tested speed.
I certainly wouldn't try Cat5e for that. Cat6a or Cat7 probably wouldn't even cut it, as even with larger conductors in there, you are likely to drop below minimum SNR over 200+m between server rooms. Not sure what gain you might get from 5e to the one jump larger conductors in 6, and twisted pair seperation...but I doubt it would be a doubling in the run (for GbE).
I'd deffinitely go fiber on this one. 200m of basic fiber should run too much, and then you have no worries about a pedestrian 200m, nor signal interferance. I don't know what fiber runs for a length that long, but I've been looking at pre-made cables for a future project extending my wired network out to a shed that I am tearing down and rebuilding as a combo dettached shed and studio/guest bedroom that is around 100-130ft from my house and about a 150ft total run from my basement server space. 150ft pre-made cables that'll support typical 1.25Gbps GBIC (which ends up being 1GbE in the end) I think were only around $60-80 when I was looking at them, plus a $40 fiber converter to host the other GBIC (my switch will take a pair of GBICs) is pretty cheap compared to trying to run 150ft of Cat5e or 6, or better yet 6a if I ever want to upgrade to 10GbE some day out there (unlikely to ever happen. I really just need a basic internet connection out there some day AND to slap in an AP to extend my Wifi network as that area of my yard is a blackhole for wireless).
Also fiber = no worries about lightning strikes (proper surge protection/lightning strike isolation seems to be a lot harder with network cables/networking gear than it is for regular old mains), and the line will be running at least 100ft buried through my yard. Plus probably being able to upgrade the speed over the fiber beyond any copper I'd possibly consider burying, some day, if I get crazy (I am eyeing up 10GbE gear, but until a 5+ port switch and a pair of 10GbE RJ45 based NICs drop below $500 total in price, I can't justify it, running a pair of 1GbE NICs in my server and desktop now with SMB3. 240MB/sec speeds are nice, but 1000-1200MB/sec would be nicer).