Having an ethernet crossover cable it is possible to make a lan, connecting both devices directly with no need of a router or switch.
For twisted-pair Ethernet, most devices support Auto MDI-X today, making crossover cables redundant.
TP crossovers were generally required for connecting similar devices directly with each other (MDI-to-MDI or MDI-X-to-MDI-X) while the standard "straight cables" were used to connect dissimilar devices (MDI-to-MDI-X). MDI devices are hosts and routers, MDI-X devices are switches and repeater hubs. Today, if any one side supports Auto MDI-X straight or crossover cable doesn't matter.
The question is: Is it possible to achieve that using fibre cable?
Yes. In contrast to twisted-pair cabling which is "straight" by standard, crossover is the rule for fiber connections. So, you can simply connect two fiber ports with a patch cable or a patch-plenum-patch run without problems (any odd-numbered sum of segments). Many fiber patch cables can also be changed to a 1:1 connection for rare situations.
Many devices have SFP (1G) or SFP+ (10G) ports that allow you to fit an optical module as required. Generally, the module types at either end must match. Some SFP+/10G modules allow you to connect to an SFP/1G module, but that is not the rule. In either case, wavelength (-S/-L) and fiber type (single-mode or multi-mode) must always match.
Does it matter the connector?
The cable connector needs to match the device connector on that side. While LC connectors are pretty much standard today, SC or even ST connectors are still widespread. Simply buy/use a cable that has the connectors you need.
What if one of them is 10GbE and the other is GbE?
Most 10G ports require you to fit a 1G module to connect to a GE port. Some port/module combinations ("dual speed") may be downward compatible (10GBASE-SR -> 1000BASE-SX; 10GBASE-LR -> 1000BASE-LX), but that depends on your equipment - check the specs. Some 10G ports won't even connect to 1G at all.
The idea is to make a direct connection between two servers using a specific fibre network card. One of them is a backup and data server we want out of our lan.
That'll work, using compatible SFP+ modules and cables. Note that some NIC vendors try to force you to use "original" modules. For almost all brands there's a large market for "100% compatible" modules though, so you don't really need to pay premium prices.
For very short range (<=7 m), a budget alternative to 10GBASE-SR is direct-attach copper (DAC) which, again, needs to be compatible with your NICs (or switches). The SFP+ modules on DACs are integrated into the cable and cannot be unplugged. With vendor lock-in, that requires you to have compatible devices at either end unless you buy or program special adapter DACs.