It will cost you around $10,000. Skip down to the last paragraph for my recommendation and then figure out whether you want to read the rest of the post.
I've been digitizing parts of my vinyl collection on and off for over 5 years and based upon my experience what you're asking is not a trivial task. It takes a lot of time and effort to properly do the task. Think for a second about all the steps that would be involved:
1. safely ship back and forth 300 albums
2. clean the records
3. play and convert to digital format
4. declick the digital files (optional)
5. manually type album name and artist
6. divide album sides into individual songs (optional)
7. manually type song titles (optional)
8. sample rate convert (optional)
9. listen to finish files for quality control
Assuming the person doing this already has the equipment there are still some ancillary costs incurred. Record cleaning fluids and materials, approximately 250 hours of wear on to the phono cartridge and a hard drive for file storage/transfer.
Basically, it comes down to for every minute of vinyl transferred it takes 2.5 minutes of work. Assuming 40 minutes per album that works out to 500 hours to complete the task. At $8/hr (minimum wage) it will cost you $4,000 plus ancillary cost (say $500). That's the absolute bare bones minimum. If you Google "vinyl to digital conversion" you'll find commercial services that will do it for $25 to $40 per album. So realistically you easily could spend $10,000 for this project.
My advice mirrors the above comment about buying used CDs. If that doesn't meet your needs then consider buying a low end phono setup for under $1,000. Such a setup would sound decent and allow you to enjoy you music. If you wanted to take it a step further, you could also buy an A/D converter and recording software and digitize the albums yourself. When finished you could then sell the phono rig and A/D for roughly what you paid. It will take much time and effort, but you will be in complete control of the process and the quality of the transfers.