Technically yes, a POE switch at the 100M (half way) should be able to make the entire link work. If there is power (power outlet) at the mid point, then you can use a regular network switch. Mind you, that depends on using good quality Cat5e or Cat6 cable. Remember, outdoor cable must be outdoor rated. Using regular indoor cable will deteriorate real quick outside, and fail very quickly.
Other solution is a wireless bridge between the two points, provided you have line of sight between the two "bridges"
EDIT: There may be other solutions where one can send Ethernet signals at reduced speeds over regular "copper wire". These adapters give long distances (1km), but not at 100mbps (but still 10mbps or better). These are called "Extenders". There are Ethernet to fiber optic cable (can go over 1km, but expensive to install), there is Ethernet to Coax (RG-6 - same as cable TV cable):
http://www.amazon.com/StarTech-com-Gigabit-Ethernet-Unmanaged-Extender/dp/B00AMCKN80/ref=sr_1_5?ie=UTF8&qid=1443212270&sr=8-5&keywords=ethernet+to+coax
which can go 2.4km
This one:
http://www.amazon.com/Kit-NV-202EKIT-HyperXtender-Ethernet-Extender/dp/B0067N3C4A/ref=sr_1_1?ie=UTF8&qid=1443212408&sr=8-1-spons&keywords=ethernet+extender
will do up to 3km over regular copper wire (again, at slightly reduced speeds, but better than nothing and faster than wireless)
Good luck