Snap has acknowledged it's "never easy to transition from receiving a service for free to paying for it" - but suggested it be "worth the cost" for users.
"These changes will allow us to continue to invest in making Memories better for our entire community," it said in a blog post announcing the move.
It said more than one trillion Memories had been saved by users since being introduced nearly a decade ago.
The feature allows users to save photos and videos shared initially for 24 hours or less on the platform, with users prompted to re-post them as a memory or "throwback" at a later date.
Users with more than 5GB of saved Memories will be prompted to upgrade to a 100GB storage plan, under the changes.
Increased levels of storage will be available to users paying for more expensive Snapchat+ and Snapchat Premium subscriptions.
The company says it will provide 12 months of temporary storage for those exceeding the limit, and users can download saved content to their device.
A company spokesperson told tech publication TechCrunch its initial 100GB storage plan would be priced at $1.99 (£1.48) per month, with 250GB included in the $3.99 (£2.96) cost of a Snapchat+ subscription.