You need to do some research about this. Some good info is available, for instance, at Murthy's web site:
http://www.murthy.com/UDirtra.html
and at
http://www.americanlaw.com/maintlpr.html
As I understand it, there are three distinct and often confused issues regarding travel abroad for GC holders:
1) maintaining versus abandoning of a PR status
2) eligibility rules for U.S. citizenship
3) documents needed for a re-entry.
There are detailed rules regarding item 2), available for example at the official INS site in their Guide to Naturalization. My understanding is that the phrase "continuous residence" is only used in the rules regarding naturalization, so it is NOT directly relevant to 1) and 3)
Regarding 3), the situation is also more or less clear:
you can use a valid plastic card (or a passport stamp) to re-enter after a less than a year absence; you need either a re-entry permit or a returning resident visa for entry after more than a year absence; moreover, the re-entry permit by virtue of its 2-year validity cannot be used for reentry after more than a two-year absence.
The most important, and most amorphous, item is 1).
In fact, merely possessing the right docoments for re-entry is not enough. Each time you re-enter the country, an INS officer has the right to refuse your admission if he/she determines that you abandoned your PR status. Theoretically this applies to both short and long trips. There are, as I understand it, no time-specific rules about this (NO one or two year rules or six months rules etc).
Rather you are supposed to maintain your intent for each of your trips to be temporary in nature and maintain your intent to resume your residence in the U.S.
This is best done by carefully documenting the temporary nature of each trip (having a definite date for planned return is good in this regard) and maintaining tangible ties in the U.S. (family, job, property, credit cards, bank accounts etc).
In your case I think that putting your stuff in storage is actually a good idea; this is certainly much better than moving it out abroad.
Maintaining credit cards, bank accounts, drivers' licence as well as filing U.S. resident income tax returns are also good things to do.
If you can sublet or sublease your apartment instead of terminating a lease, this may also be a good idea.
The good news is that INS agents rarely bother asking any questions, especially if you have a valid reentry permit. Moreover, as I understand, they generally consider family circumstances (such as looking after a sick/old/young relative) to be a satisfactory reason for an extended temporary absence from the U.S. You should, of course, try to carefully document this, just in case.