Just throwing this out for consideration.
You say that you have never been in any kind of trouble with the law, so...are you sure it is a CORI matter?
Is it possible that DCF or local probate court probation officer or investigator was doing a background check on you for the purpose of child custody and found something? If they are holding a past bad act against you, you have a right to know what it is and the right to refute it. Some enemy or wrong headed person might have told lies about you that the investigator wrote down. Sometimes things get said and it becomes impossible to get it removed once its on paper.

http://www.mass.gov/courts/selfhelp/criminal-law/seal-record.html is the site to go to if you want to look into sealing your record.

It sounds as if the problem you are talking about may not be a CORI problem but rather a problem with the Department of Children and Families. If an allegation was ever made against you that you neglected or abused a child and if that allegation, after an investigation, was supported by the Department of Children and Families (DCF) then your name is on a register (list) maintained by DCF of persons who neglect or abuse children. Schools, day-care centers, adoption agencies, and the like have access to this list. This may be what is causing you a problem if it is something that is not on your CORI.

I would suggest finding a criminal law attorney using AVVO to clear your CORI for you. Ask them if they help clear CORI records---some do not. The criminal attorney can walk you through getting the CORI, checking what us incorrect, and filing the motion and argument in court to clear the CORI. The attorney can also explain what a CORI is to you and how it affects your background.

The other attorneys here explained how you can get a copy of your CORI. It have something that is at error on your CORI, you can have it removed. It depends, however, on what the error is. I'll send you a link from MASS Legal Help that shows you how to have errors or mistakes on your CORI removed, rather easily.
Next thing, please do not worry if you are gay. Most people nowadays really believe that gay people are very cool, so no need to worry about that. If this continues to haunt you or worry you, it's a great idea to consult a professional who can help you with this.

The first step is to request a copy of your CORI. You can do this online if you have a valid driver's license or by mail. Your CORI will show the court, charge, arraignment date, and disposition of the case. Once you have those details then you can determine what information is wrong. Your CORI could list a charge that you never faced or it could show the wrong disposition (i.e. Guilty instead of CWOF). There are several avenues you can pursue to remedy an incorrect CORI. A few of those options include: contacting the probation department at the courthouse where the case occurred but they can only fix minor mistakes or clerical errors. You can also contact the Office of the Commissioner of Probation at 617-727-5300. OCP can be helpful depending on the mistake. Another option is to file a motion with the court to fix your CORI. There is also a form you can fill out and submit detailing the incorrect information. The avenue you pursue ultimately depends on what the error is. I would recommend you contact an attorney once you receive your CORI because they can advise you on the best way to fix it. Unfortunately fixing errors on your CORI can take some time and it can be frustrating. Hiring an attorney that has experience in fixing CORIs could be very helpful to you.

You can order an electronic copy of your own CORI from the state's iCORI website. It will cost you $25. I've included a link to the website.

If your identity was stolen or if the government intended to charge Person A but mistaken.y charged you instead, you could have the charge expunged from your CORI. Absent these two circumstances, you cannot get an expungement and your next best remedy is to seal your CORI.

Sealing your CORI will make your CORI and the courthouse files unavailable to most parties who request to see your criminal history. Law enforcement agencies can still see your CORI post-sealing, as can the Department of Children and Families in when you're seeking to adopt a child.

I suggest that you speak to an attorney for an analysis of what options are available to you in your situation, and the best strategy for achieving your desired outcome.

Best of luck,
Dominic Pang

I could not be sure exactly what is on your CORI without looking at it but here is a link with general information:

http://www.gbls.org/sites/default/files/GBLS_criminal_record_sealing_and_cori_reform.pdf

See the link below, which I got to by googling "How can I find out what is on my Cori" from your question.

Once you find out what's there, you can go to the court that it came out of to correct any errors.

I changed the category of this question so that attorneys most familiar with CORI processes and correcting criminal record errors might see it.