So, I think I can conclude what's wrong with the circuit in an answer after having discussed it in the comments:
- You say you need to minimize current consumption, yet you burn 13V (24 V - 11V) in your linear power supply. Don't do that. Use a switch mode power supply (step-down) to do e.g. 24 V -> 12 V (effectively halving the power consumption on the 24V rail) and do the rest (12 V -> 11V) using a LDO (low drop-out) regulator, if you need the exactness. You probably don't – opamps do have a supply voltage rejection that should eliminate the effect of supply voltage uncertainty on your signal.
- D2 is an especially bad idea since it must lead to additional current being wasted, at least in "ON" state. Also, as Andy pointed out: 9V is even higher than what the LMX358 is specified for! So, this is even twice a wrong choice.
- if you want to reduce current draw, use MOSFETs instead of bipolar junction transistors
- A simple high-side MOSFET switch would do the job without you burning energy (and thus, 24V current) on the forward voltage of your 1970s BJTs as well as their base currents
- LMX358 still has a relatively high quiescent current and no disable/enable pin. You could go to TI.com, and click through to "amplifiers -> Opamps -> Ultra low-power opamps", and pick one that directly works off the voltages you have, has low quiescent current, and a high supply voltage rejection (so you can directly connect it to your SMPS instead of the LDO). In the best case, it would also have an enable pin, so you can simply shut it down when you don't need it. In your case, the simplest solution would be to replace the LMX358 by an LP358, which isn't even more expensive, usually.
- I highly doubt your requirements for low current if you have a linear supply, and some unnamed microcontroller that you use, as well as your 30 Ω R4. Also, 24V doesn't come from "anywhere". If this is from a lead-acid battery, or an array of alkaline batteries, or an array of NiCd/NiMH rechargables, compare your currents to the self-discharge currents. You'd be surprised. If this is coming from a grid (220V/110V...) adapter, then ignore the quiescent current, since the conversion efficiency for small loads will be terrible, anyway. If this comes from a solar cell, same applies.
Furthermore, since this is an analog signal processing circuit, there's a distinct lack of decoupling capacitors (and a distinct lack of consideration for the current these leak).
As usual, pick the right components for your job based on what they need to do. If you can restrict the bandwidth you need from the opamp to a couple kHz and the gain to a couple V/V, you can easily drop in a nA-quiescent-current opamp.