Do it in stages, beagle drives a FET which drives the relay. The FET
is low-side (source connected to ground) and the high side of the
relay's control goes to a 24V (or similar) source. The relay can
switch your system's actual power (the sprinkler siphon valve or
what-have-you) at what ever voltage that needs. Just pick a relay that
can handle switching your load on-off-on.
If you're able to use 12 V relays, your local auto-parts store has a
bunch in stock that can handle quite a lot. Usually they're pretty
If you tell more about what your components are, such as links to
specs (with voltage, current, etc documentation in them), we'll be
better able to help.
The Beagleboard will have enough power to drive a FET, which will power a relay, which in turn will power the valve. If you do not use a solid state relay, put a reverse biased diode across the coil to protect the FET from any back EMF. Also it would be prudent to put some transient protection/noise suppression on the relay-valve circuit. Just my 2 cents.
Yes, exactly this.
A relay like  would probably work.
What you'll need is a 24 V AC source, as that's what your sprinkler
says it needs. You use that 24 V AC source going to the solid state
relay which acts like a light switch (you can use an actual light
switch to prototype, if you want).
The low voltage control input to the solid state relay will need to
have +5 V on one side of the solid state relay input and a FET on the
other, with the FET source connected to ground and the FET drain
connected to the solid state relay. You'll want an N channel FET with
a gate voltage less than 1.5 V if you're using an xM (you could get
away with a 2.5 V FET gate with a bone).
The xM drives a GPIO which turns on and off the FET. The FET turning
on and off turns on and off the relay. The relay turning on and off
turns the sprinkler on and off. Due to the difference in power and
voltage levels required, you have to do this in stages.
There are many ways of doing this. Personally I would use suitable MOVs ) across relay contacts that drive inductive loads. Also make sure that the ground currents cannot flow through any sensitive circuits; a single point star configuration is what I have used in the past.