To solve the given optimization problem using Gurobi, we first need to translate the natural language description into a mathematical formulation. The objective is to minimize the function $6x_0x_1 + 8x_1$, where $x_0$ represents the number of '3D printers' and $x_1$ represents the quantity of 'packs of paper'. The constraints are as follows:

1. Dollar cost constraint for 3D printers: $6x_0 \geq 33$
2. Weight constraint for 3D printers: $20x_0$ contributes to total weight
3. Cost constraint for packs of paper: $9x_1$ contributes to total cost
4. Weight constraint for packs of paper: $9x_1$ contributes to total weight
5. Minimum combined spend on 3D printers and packs of paper: $6x_0 + 9x_1 \geq 33$
6. Minimum combined weight of 3D printers and packs of paper: $20x_0 + 9x_1 \geq 84$
7. Special constraint: $x_0 - 4x_1 \geq 0$
8. Maximum spend on 3D printers and packs of paper: $6x_0 + 9x_1 \leq 70$
9. Maximum combined weight of 3D printers and packs of paper: $20x_0 + 9x_1 \leq 168$
10. Integer constraints for both variables: $x_0, x_1 \in \mathbb{Z}$

Given these constraints, the problem can be formulated as a Mixed-Integer Linear Programming (MILP) problem. However, note that the objective function is not linear due to the term $6x_0x_1$. For simplicity and to adhere to standard MILP formulations where Gurobi excels, we'll treat the objective function directly but keep in mind this introduces a quadratic element which requires careful handling.

Here's how you could implement this problem using Gurobi in Python. Note that since our primary goal is minimization under the given constraints, and considering the complexity introduced by the non-linear term in the objective function, we will simplify our approach to fit within the linear programming framework where possible or adjust to handle quadratic terms directly.

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x0 = m.addVar(vtype=GRB.INTEGER, name="3D_printers")
x1 = m.addVar(vtype=GRB.INTEGER, name="packs_of_paper")

# Objective function (Note: This is a simplification to fit into the linear framework)
m.setObjective(6*x0*x1 + 8*x1, GRB.MINIMIZE)

# Constraints
m.addConstr(6*x0 >= 33, "Minimum_3D_Printers_Cost")
m.addConstr(20*x0 + 9*x1 >= 84, "Minimum_Combined_Weight")
m.addConstr(x0 - 4*x1 >= 0, "Special_Constraint")
m.addConstr(6*x0 + 9*x1 >= 33, "Minimum_Combined_Spend")
m.addConstr(6*x0 + 9*x1 <= 70, "Maximum_Combined_Spend")
m.addConstr(20*x0 + 9*x1 <= 168, "Maximum_Combined_Weight")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"3D printers: {x0.x}")
    print(f"Packs of paper: {x1.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```

Note that this formulation may not perfectly capture the quadratic nature of the objective function as described. For a more accurate representation, you would need to use Gurobi's capabilities for quadratic programming (QP) or mixed-integer quadratic programming (MIQP), adjusting the model accordingly.

To handle the quadratic term directly in Gurobi, you might consider using `m.addQConstr` instead of simplifying the objective function into a linear form. However, this requires ensuring that your problem is correctly formulated as a quadratic program and that all constraints are properly represented within this framework. 

```python
# Example of how to add a quadratic constraint (not directly used here but relevant for handling quadratic terms)
# m.addQConstr(6*x0*x1 + 8*x1 <= some_limit, "Quadratic_Constraint")
```

Keep in mind that the correct formulation and solving of optimization problems depend heavily on the specific details of your problem. Always ensure to validate your model against the original problem description to guarantee accuracy and relevance of the solution obtained.