Overcome this situation is substitute x=1/t^2 then plot and integral is:
Plot[{(2 E^(-(1/t^2)))/(
t (BesselJ[0, 1/t]^2 + BesselY[0, 1/t]^2))}, {t, 0, 20},
PlotRange -> All]
NIntegrate[(2 E^(-(1/t^2)))/(
t (BesselJ[0, 1/t]^2 + BesselY[0, 1/t]^2)), {t, 0, Infinity},
Method -> "GlobalAdaptive", WorkingPrecision -> 100]
(*4.85471500790454846564370835036737574863704130040306375617123990141541\
1155533681789950223687667641287*)
NIntegrate[(2 E^(-(1/t^2)))/(
t (BesselJ[0, 1/t]^2 + BesselY[0, 1/t]^2)), {t, 0, Infinity},
Method -> "GaussKronrodRule", MaxRecursion -> 40,
WorkingPrecision -> 100]
(*4.85471500790454846564370835036737574863704130040306375617123990141541\
1155533681789950223687667641287*)