How to protect my digital data from Solar Flares?
The problem with a geomagnetic storm is that it will induce currents in conducting objects, such as power lines.
A geomagnetic storm shares some properties with, but it is not, an EMP. Or better: what we think of as an EMP is a very localized phenomenon, which at relatively close range has energies orders of magnitude above the ones of a CMU that happened 150 million kilometers away from the Earth.
So an EMP like STARFISH PRIME's would trash your equipment if it happened within a few (tens of?) kilometers, depending on shielding. A geomagnetic storm will induce currents that are proportional to the length of the conductor coil. While this is very bad if you're a power station connected to a 500 km set of wires, it is next to harmless if you're a home PC no wider than a few tens of inches.
But telluric current protection is in effect on most power lines since the bad period in the 1990's, and hard disks and the like are lightly shielded against everyday magnetic interference.
Only the severest magnetic storms are able to pierce Earth's magnetosphere, in which your hard disk lives immersed. It follows that only rarely a magnetic storm will develop local strengths of more than, say, five times Earth's magnetic field, which is around 0.50 Gauss.
And hard disks (and consumer electronic equipment in general) are largely immune to fields up to six hundred times that.
For example, in RAID disks, you'll have DISK 2 which is spinning very close to disks 1 and 3, which come equipped with very powerful rare earth magnets. From 2's point of views, those two magnets are an interference and a harassment, yet RAID disks perform flawlessly for years.
Additionally, desktop PC are usually encased in a steel or iron case which is not only antistatic (a Faraday cage) since it is metallic, but it is also antimagnetic since it's made of ferrous alloy. Laptop PCs have lighter alloy cases (from most to least expensive, titanium, magnesium, aluminum and plastic) which are not antimagnetic (plastic is not even antistatic - or not very much even when the inside is surface treated with conductive paint).
However, there are magnetic shielding cloths that will increase your equipment's resistance to solar flares by anywhere from 2 to 7 orders of magnitude; I remember seeing an antistatic/antimagnetic/RFID-proof case for Mac Air on Amazon, so I'm pretty sure they should exist for other models too.
You have time to prepare
As you can read in the Wikipedia article about the Carrington event, 2 days before the solar storm, a lot of sunspots were detected. This, in combination with warnings from the NOAA forecast of M and X GOES class solar flare events give you a bit of time to prepare.
Events much less severe as the Carrington event
There are two basic strategies here:
As you can see in the EMP simulator, not the entire earth is directly affected by a solar flare. So distributing your data over the globe is a good strategy. Several cloud storage services deliver such services.
Grounding and shielding your electronic equipment could shield your data by diverting the EMP to the ground instead of meshing up your data. Shielding is best achieved with a Faraday cage. Wrap your electronics in several layers of well conducting material (aluminium foil, tin litter can, etc.), separated by non-conducting materials (plastic or PVC will do fine). In the most optimal case, ground the outer layer with the ground water or a large body of water, so the cage can discharge. A more elaborate description can be found here.
Events similar or bigger than the Carrington event
The best solution is to store your data in non-electronic format. CD/DVD's should work fine (source: The Official CHFI Study Guide). If you want something more safe than regular or archive quality CD's/DVD's, you could put all your data on an M-Disk.
M-DISC is an archival-quality storage solution that preserves photos, videos, music, and documents for 1,000 years or more. Unlike hard drives, flash drives, and other writable media, that can lose data, M-DISC has been designed to protect your information from degradation and loss for centuries.
Also printing your documents and photo's is a way to keep a non-electronic copy. After the cataclysm, you could re-digitize your photo's and documents.
As Lloyds predicts, the impact will be severe, and electricity will be down for a long time (the Lloyds report mentions weeks and even months). Without power, daily life will soon become chaotic and your lost photo's is probably the least of your worries.
Theoretically, if your Faraday cage is good enough (enough layers, thick enough conducting layers), it might pull it off. But I'm not sure if that will work with such violent events.
It would depend on the size of the event. The event causes a huge magnet wave, which could induce current to flow in any electrical carriers. That means the power grid, phone lines etc. But it would also de-gauss magnetic storage if strong enough. A faraday cage under your basement is likely the best option for an individual.
For extreme protection you would build a shielded bunker and place it near to the equator where the wave would be felt the least.
Realistically though you wouldn't care. No power, no water, no communications. Food shortages after a few months. Losing a few JPGs would be the least of your worries.
NB: Physics/Electrical Engineering could give a much more technical answer than my laymans summation here.