I had all of my games installed on my xbox 360 hard drive, but then my brother told that hard drive is getting too full, so I deleted the games off of xbox, and then I lost all of my game progress. All the profiles on my xbox lost game progress. Is there anything I can do to get game progress back.

This is very simple: you seem to not have enough hard drive space for a system update that your console is trying to install.

Error 8007 0070 occurs when you install an update or buy content on...

(I've pasted some bullet points for you)

" Error 8007-0070 occurs when you install an update or buy content on Xbox 360
Problem
You may see the following error code and message when you try to update your Xbox 360 console or purchase content:
xxxx-xxxx-xxxx-xxxx-8007-0070

Note The "x" characters in the error code vary.
This may mean that there's not enough space on your storage device (mostly likely the Xbox 360 hard disk) to install the update or the content.

Solution
A console update or content download may require significant hard disk space. If you don't have enough free space, you need to move or delete some items."


You can probably get around this by using a 32gb USB thumb drive and moving some of your content that's hogging up your console's space onto that thumb drive temporarily.

Heres the Xbox.com info to get you started about moving and deleting hdd space--which also has info about using a USB thumb drive, too. Copy move or delete Xbox saved games profiles and avatar items

You may have an old series console version of the Xbox 360 that only has either 256MB or 512MB of internal memory. You may also have a new series console version of the Xbox 360 that only has 4GB or data storage. For multi-disk games that install one or more disks, you need, at minimum, a 20GB hard drive installed in your console. You will need to invest in a hard drive for your Xbox 360.

It probably wants you to install a memory card to get full potential out of your gaming system, and required in more complex sofware setups.

It is possible that your game is scratched up enough that you can't play it directly on your xbox 360... BUT you might still be able to play it is you can save it to your Xbox 360 hard drive. You might need to try it a few times to get it to read and work but try following these instructions for installing the game from your CD directly onto your Hard drive. You will still need to have the CD in your Xbox 360 to play but it will be reading from your hard drive if you can install it. Good luck:

To install a game on your hard drive loadTOCNode(2, 'moreinformation');

1. Insert the game disc into the disc drive.
   
   
   
   Note After you insert the disc, the default option is to automatically start the game. If this happens, exit the game and return to the Xbox Dashboard.
2. From the Xbox Dashboard, go to My Xbox.
3. Select the game you want to install.
4. Press the Y button on your controller.
5. Select Install to Hard Drive.
6. A progress indicator appears while your game is being installed on your hard drive.

Note Installing an Xbox 360 game might take 12 minutes.
To play a game from your hard drive loadTOCNode(2, 'moreinformation');

1. Insert the game disc into the disc drive.
2. From the Xbox Dashboard, go to My Xbox.
3. Select the name of the game you want to play.

Note The game disc must be in the disc drive, even though you have installed the game on your hard drive. The console will automatically play the game from your hard drive, but the disc has to be present.
To delete a single game from your hard drive loadTOCNode(2, 'moreinformation');

1. Insert the game disc for the game you want to delete from your hard drive.
2. Go to My Xbox, and then select the game you want to delete.
3. Press the Y button on your controller.
4. Select Delete from Hard Drive.

To delete multiple games from your hard drive loadTOCNode(2, 'moreinformation');

1. On your console, go to My Xbox, and then select System Settings.
2. Select Memory.
3. Select Hard Drive.
4. Select Games.
5. Select the game you want to delete.
6. Select Delete.

To clear space on your hard drive loadTOCNode(2, 'moreinformation'); If there is insufficient space on your Xbox 360 hard drive, you will receive the following message:
Not enough space to install this game. It needs [space needed] free space to install. Do you want to clear space? (YES/NO) Click YES to view a list of games that you can delete. After you clear enough space on your hard drive, you can install the new game.

I have never heard this issue before. You probably have a lager issue then what can be resolved here and might face future issue in the future. Is this the only game you are having issues with?

Seeing that you mentioned you have purchased a hard drive then I would recommend you install the game to your hard drive and try play after that. This will resolve any issues possably of your disk drive possably getting hung up during gameplay.

follow these instructions to get your game installed on your hard drive:

Install Xbox 360 games on your hard drive from a game disc Note Installing an Xbox 360 game on a hard drive requires between 4 and 8 gigabytes (GB) of free disc space.

1. Insert the game disc into the disc drive.
   
   
   
   Note After you insert the disc, the default option is to automatically start the game. If the game starts, exit the game, and return to the Xbox Dashboard.


2. From the Xbox Dashboard, go to My Xbox.
3. Select the game you want to install.
4. Press Y on your controller.
5. Select Install to Hard Drive.
   
   
   
   A progress indicator appears while your game is being installed on your hard drive.

Notes

• Installing an Xbox 360 game might take several minutes.
• Multidisc game? Install each disc to your hard drive.

if you wana say that you installed a game on your xbox elite and run on different profile.it works fine
but as you used a differenr xbox but older hdd(elite hard drive).your profile runs but installed games not.
BECAUSE THAT GAME IS NOT ORIGNALY INSTALLED ON THAT DVD DRIVE,you are using.delet it and installed again.onlt game not save files
it will use your save files.

Reformat the hard drive using the Xbox 360 system. If this does not work, then it is time to get a new hard drive for your system. We sell 20GB hard drives for the Xbox 360 so you can contact us and we'll be glad in helping you.

- You've been helped by a Wired Extreme technician.

If its a modded console, then you need to update your firmware to the newest version.

If its not modded, check the disc for scratches/dirt. Also you can try to delete the local data for that game on your xbox 360 hard drive.

Remember when you do this delete, you lose all downoaded game updates and you need to re-install any you use.

1. Hold the "A" button while turning the Xbox 360 console on.
2. Get into the system menu and select "Memory" from the options.
3. Select the hard drive, and press "Y" to get the device options screen to appear. See a "Rename" and "Delete" options, but don't choose either.
4. Press the following code in: "X, X, left bumper, right bumper, X and X"
5. View a message that your Xbox 360 has performed system maintenance. Select "Yes" to confirm you want to delet the cache.

If that still doesnt work, your optical drive may be going bad and having trouble reading.

there are only a few games for the 360 that require a hard drive in the first place (but without one you wont be able to save your game).
if you have one, the only possibility i can think of is maybe the hard drive is full, in which case you just need to delete a few things off of it. most games dont need very much space.

note: I'm not sure what you mean by new games and regular games. so if this doesn't help, make another post clarifying and ill see what i can do.

Yes, it either doesnt work, or the game your using isnt compatible period.