Q:

How is it possible for manufacturers to constantly get data from their planes all over the world?

I've heard that manufacturers (like Boeing and Airbus) constantly get data from their planes. So in the case of an emergency, ATC's or pilots can contact the manufacturers for assistance. How do manufacturers constantly get this data and how is it stored (cause considering the number of planes flying today, it must be a lot)?

A:

How do makers get data from aircraft in flight
The data is usually transmitted using something like ACARS which is a system used in aircraft to transmit data in flight using underlying radio and satellite communications systems.
The communications infrastructure is typically provided by large multinational companies who specialise in this job. See ARINC, SITA, Inmarsat and so on.
Engine-makers and Aircraft makers often offer pro-active maintenance and support services to airlines, operators and lessors which involve their aircraft transmitting such data to the makers.
For example, Airbus has AIRTHM

Airbus Real Time Health Monitoring (AiRTHM) is an advanced service through which A380 and A350 XWB operators can receive guidance on optimised maintenance and real-time troubleshooting actions. As part of this effort, a dedicated 24/7 team takes advantage of the uplink technology to further investigate and anticipate warnings/fault consequences.
The uplink technology allows real-time remote access to aircraft data parameters via the ACARS digital datalink system, enabling Airbus AiRTHM engineers to deliver maintenance and technical advice both in flight and on the ground.

Boeing have AHM

AHM sutomatically monitors, collects and transmits service levels using ACARS through the installed Aircraft Condition Monitoring System (ACMS)

Note that ACARS is a relatively low-volume data transmission system. Makers have explored other underlying communication systems as a way to transmit more data at lower cost.
How is this data stored
It is true that there is a lot of data but Boeing and Airbus are big businesses and are able to partner with major IT businesses.
For example IBM work with Airbus

“Today’s aircraft can generate up to a half terabyte of data per flight, an unprecedented volume and variety of data seen in few other industries,” said Timothy Wholey, IBM global leader for the aerospace and defense industry in its global business services unit.

Note - not all data has to be transmitted in flight. A lot of data is collected and stored in the plane and only transferred when the aircraft is on the ground.
Makers like Airbus and Boeing inevitably are already very heavy users of IT for aircraft design purposes. For example, Airbus use HP Supercomputing hardware

