Are you running macOS 10.12 Sierra as your signature says?

I'm not sure I understand "150 gb of space on my mac and 98 gb is taken up by Photos" A MacBook Air comes with a bigger hard drive than that.
You say you are using iCloud-- do you have iCloud turned on in Photos? Do you have "Optimize Storage" turned on in Photos>Settings>iCloud? If you don't have that set to "Optimize," then turning that on can save lots of space. The disadvantage is that you can't make a robust backup of your Photos Library.
AlWeir's suggestion of using an external drive for your Library is a good idea. An SSD is faster than a mechanical drive, and that's important, but it's also more expensive. As suggested, after you get a new drive, it should be formatted for for the Mac. The biggest problem with it is that an external drive really needs to be kept plugged in, and that's problematic on a laptop like your MacAir. Photos does lots of background stuff, and unplugging the Library can be confusing to it, especially if you're using iCloud. If you choose this route, let us know and we may be able to give some suggestions. You should see this:
Move your Photos library to save space on your Mac - Apple Support