It sounds like the aperture diaphragm in the lens is stuck. If it is stopped down past f/5.6 during focusing the camera's phase detection auto focus will not be able to work properly. Normally focusing is done with the lens wide open regardless of the selected aperture. The lens isn't actually stopped down to the selected setting until just before the shutter opens. If you select Manual on the mode dial you should be able to focus manually and fire the shutter whether the lens is properly focused or not. Since f/5.6 is the largest aperture the EF 75-300mm f/4-5.6 is capable of at 300mm, if the aperture is stuck even a little bit stopped down the camera will have problems achieving focus confirmation that is required before the camera will fire in most shooting modes.
If the steps below are not successful, or even if they are, I recommend replacing the EF 75-300mm f/4-5.6 with the EF-S 55-250mm f/4-5.6 IS. The 55-250 is a better lens optically than the 75-300 (and pretty much the optical equal of the more expensive EF 70-300mm f/4-5.6 IS) and because it is an APS-C only lens it is also smaller, lighter, and almost as inexpensive as the 75-300. It also will give you Image Stabilization.
There are a few things you can try:
Set the camera to Manual shooting mode (M on the Mode Dial). Select an aperture value slightly narrower than what the camera is using when you shoot in Live View. (If the camera says you are using f/8, try f/9 or f/10, If the camera says f/7.1, try f/8.)
Look through the viewfinder and press the Depth of Field (DoF) Preview button to stop the lens down while still looking through the viewfinder. If the view becomes slightly darker when the DoF Preview is pressed, then the aperture servo is moving it in that direction. If the aperture darkens, proceed to step 3, otherwise proceed to step 2.
If the view does not darken noticeably, then try a higher f-number. By the time you have set the f-number value one full stop higher than the what the camera is using in Live View, there should be a noticeable difference in brightness through the viewfinder. If so, proceed to step 3. If the view does not darken regardless of the aperture setting, the servo that moves the aperture diaphragm is probably defective. Go to step 5.
If the view darkens when the DoF Preview is pressed, release the button while still looking through the viewfinder. If the view gets lighter, then the aperture servo is also moving it in the opposite direction. Proceed to step 4. If the view stays darker, then the aperture servo is probably defective. Go to step 5.
Select the narrowest aperture (highest f-number) available for your lens. Press and release the DoF Preview button several times. Try holding the camera at several different angles. The idea is that if there is something physically preventing the diaphragm from fully opening it will be dislodged. If the problem persists, go to step 5.
If the lens is till under warranty, contact the Canon Service Center for your area and follow their instructions regarding warranty service. If the lens is no longer under warranty, proceed to step 6.
At this point you don't have a lot left to lose. The cost of shipping the lens to Canon and having it cleaned and calibrated (not to mention repaired) outside of warranty is almost as much if not more than what you probably paid for the EF 75-300mm f/4-5.6. Set the aperture to the narrowest value (highest f-number) available. While holding down the DoF Preview button with the camera powered on remove the lens from the camera. Place a body cap or other lens on your camera, set the camera aside, and place the back cap on the lens. Lay a fairly thick towel on a firm surface and Gently tap the side of the lens on it. Rotate the lens about 1/8 of the way around and tap the side of it again. Continue this until you have rotated the lens back to the where you started. Mount the lens back on the camera and test again. If the problem persists, go to step 7.
You can: a) Find a tall building, get a friend to stay on the ground outside and videotape while you go to the top and drop the lens off the side (Please be sure no one is directly below you!). b) Try to sell the lens "as is" and "for parts only" on eBay, craigslist, or another forum (Good luck with that!) c) Buy a reversing ring that will fit the threads of the EF 75-300mm f/4-5.6 and try your hand at Macro photography d) Add another paperweight to your collection.