How could I get rid of my bulgy eyes? (Photo)

I would consider a small surgery where I release the lower eyelid retractors.  This allows the eyelid to naturally elevate.  The release is done from a small incision on the inside of the eyelid.  What this does is it creates more a space between the muscles and their attachments.  I have seen this work beautifully in many young patients.  You could expect about 2mm of lift.  This would be enough to make the eyes look less buldging.

You have genetically big appearing eyes.  The lack of volume in the lower lid accentuates this appearance.  If you use some filler here you should notice an improvement.  It will decrease the dark circles and help to move the lower lid up so you don't see as much white under the eye.  The HA fillers such as Belotero or restylane would be good for this.  If you like how it looks,  you could then do fat. The more permanent solution is an orbital decompression.   

Combination of orbital decompression and filling around the eyes (either with fat or fillers) will give the best result. That doesn't mean that doing only one of the procedures above won't be beneficial, just less so.

If you know for sure that Grave's disease is not the cause, then the best option is to add volume to fill in the areas around the eyes.  This is frequently done to treat signs of aging and options that can be used include fillers (such as Restylane, Juvederm, or Belotero), fat (harvested from your abdomen or thighs and then placed around the eyes), or surgical options to reposition the tissue around the eyes.  Each option has its pros and cons and what is the best options depends on the individual patient.Good luck!

The photgraph you show here appears more as periorbital hollowing than exophthalmos. Fat micrografting to the upper and lower orbital bone rims will help considerably in my opinion.  While off the shelf fillers may work this is an off-label use and the results are somewhat unpredictable. A thorough facial analysis by a board certified plastic surgeon is in order.

Orbital decompression is a complicated surgery for a patient with no pathologic diagnosis.  Camouflaging the appearance by adding volume, particularly in the upper lid area may be the best approach.  This could be done with either an injectable such as hyaluronic acid or fat.  Seek out a board certified surgeon in your area with experience in such procedures.

You only show the right eye which makes it impossible to judge the symmetry you seek.  Nevertheless, the bulging side typically shows more "white" than the opposite side.  A simple and reversible way to mask this difference is with limited tarsorrhapies (sewing the upper and lower eyelids together) at the inner and outer corners.  This will reduce the visible scleral triangles and can be titrated to match the opposite side.  If you don't like the look, it can be simply reversed.

Lack of projection of the infra-orbital rim may be corrected with fillers.Hyaluronic acid fillers are a good first choice as they are reversible. Fat and Artefill are more long lasting options.Surgical intervention with advancement of the cheek is a more aggressive and permanent option reserved for more severe cases.Please consult in person with an experienced Board Certified Plastic Surgeon prior to making treatment decisions. Cheers