Hi. Yes you can.
Check this component out. It costs under £3 including postage and may be just what you are looking for.
http://www.ebay.co.uk/itm/140931469538
It has a detection range of about 7 metres and works from a power source of between 5v and 20v DC, so for example, you could use a 9v battery to power it. Better still, you could also use an old 12v car battery to power it (even if the old car battery is no longer fit for purpose for starting your car). Old car batteries have plenty of punch even when they are no longer suitable for starting cars.
You can vary the detection distance from about 3m to 7m, and also vary the time the alarm sounds, from 5 seconds to 300 seconds. There are two little dials (referred to as potentiometers) that you can turn to the settings you prefer.
Plus it seems you can add a photo-resistor (looks as though it already has one on the circuit board, but anyways)...to disable it during daylight.
The output when triggered by an intruder is 3.3v, enough to sound an alarm through a small speaker or to trigger a much more powerful speaker if you were to use a switching power transistor (for example you could use an N-Channel Mosfet as a switch [a switch that turns on when it receives the 3.3v signal] to turn on a more powerful device). I am still learning about Mosfets so please dont ask me any complicated questions about them until I have time to learn a little more
I would make a plan, including a circuit diagram, of what you intend to build.
Personally, I would enjoy installing a small solar panel on the shed roof, and then connect the two cables coming from the solar panel (a red +ve cable and a black -ve cable) to a solar charge regulator. Solar charge regulators are well labelled so you can easily tell which way round the cables from the solar panel should be connected to it.
Then you also connect the positive and negative car battery connections to the solar charge regulator (taking care to get the polarity right or you may melt the circuits in your solar charge regulator.
Finally, you connect the load or output terminals of the solar charge regulator (the terminals on the right hand side of the solar regulator in the photo below...via the 12 volt regulator though it is optional) to the intruder detector power input terminals.
The solar charge regulator prevents the old car battery getting damaged as a result of being overcharged or over-discharged.
So for sure I would use an old car battery as my battery source. The solar panel will ensure the battery always has more than enough charge (even if you only get a little 10w solar panel), and indeed you will almost certainly have enough additional power to light the shed using LED bulbs as well from your solar power supply.
I would maybe use an L7812CV (or similar) voltage regulator integrated circuit to ensure a smooth 12v DC power supply output to the intruder detector device (it is a three pin voltage regulator that guarantees a nice smooth voltage supply when you add a couple of capacitors to it...and then...relax in the confident knowledge the alarm has a free source of power (if you ignore capital expenditure) almost indefinitely, and is protecting itself as well as the shed from intruders.
Then decide what speaker or siren you want it to trigger. The options are endless. Build it indoors and test it before installing it on the shed.
I posted photos of a solar charge regulator and some L7812CV voltage regulators so you can see what they look like.
I don't even need one of these little intruder detection devices but I was so impressed by it I added to my Ebay basket last month anyway