Even if your storage for resources is full you will still get the extra resources from an offer purchase. Any offer purchases made that include resources such as food, wood and stone will ignore the storage cap and add the resources to your account. You will be unable to exceed the storage limit in other ways (besides upgrades and research for example) but any purchase of an offer will always ignore the storage limit.
If your equipment inventory is full you will be unable to exceed this storage limit, regardless of if an offer was purchased or not. There are two limits to the equipment inventory, a soft limit which will change the look of the inventory to include a large, red exclamation mark and the hard limit. The soft limit is in place to give you an idea of when your storage is almost full and allow you to remove some items if rewards are expected. The hard limit on the other hand cannot be exceeded, either with purchases or gaining rewards from events or battles. Please be sure you have enough space for any equipment, gems or heroes you expect to receive.