I was thinking about things like wine or something Jordan.J.D had left in comment, until I saw jmb.mage's comment under rand al'thor's answer and quickly came up with something else.
I guess you are
Hard disk drive
I was round a long time ago.
This may refers to the IBM 350 disk storage, a component of IBM 305 RAMAC, the first hard drive disk being used in commercial computer for the secondary storage in 1956.
Here's a picture of the drive:
Then I was brought to you without breath and you got upset when I went around the bend saying I was no longer fresh enough.
(Still puzzled with this line...)
I used to come and go, but now I decided to stay, or at least it seems that way.
It supposed to be disk packs, a removable data storage. Modern HDD platters are fixed inside the device.
I am made up of many, but you never notice and think I am just one.
Usually a hard disk drive is composed of more than one single platter, yet we may think only one "disk" was installed.
Oh boy, the older I get, I keep getting smaller.
As we kept using the same hard disk drive, spaces available for data storage could gradually shrinking.
Sometimes I tear but not rip, and it is not my fault.
This refers to the partitioning on a HDD, which is made nicely through the operating system that ordered by the user, rather than done by itself.
In case I was dead, you'd probably keep me around anyway.
Even a HDD had crashed, we will call for data recovery - especially a broken device with some important data inside!